The Mathematical InteUigencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
A New Heisenberg PrincipIe One increasingly witnesses young mathematicians struggling to orient themselves in the mathematical world of their hand-waving established elders. It is a world of "well-known" and "obvious" concepts and results, often attributed, if at all, to sources where they are not to be found in any clear statement. Some persist in the frustrating attempt to resolve what is really meant and what is really valid; many, unfortunately, quickly adopt these fast and loose ways with fact and speculation. A recent spate of articles suggests that some people are becoming conscious of something seriously amiss. See for instance " 'Theoretical Mathematics': Toward a cultural synthesis of mathematics and theoretical physics," A. Jaffe & E Quinn, Bulletin of the American Mathematical Society 2 (1993), 1-13; "Theorems for a price: tomorrow's semi-rigorous mathematical culture," Doron Zeilberger, Notices of the American Mathematical Society 40 (1993), 978981; "The death of proof," John Horgan, Scientific American, October 1993. It may be that we are confronted with something far more fundamental than a mere cultural convergence of mathematics and physics:
Topologically this link is the Borromean R i n g s - - a link named after an Italian family of the Renaissance period who had a pattern of three circles interlinked in such a way on their crest. John Robinson has used this motif in two other sculptures: "Cxeation" and "Genesis" are Borromean squares and diamonds respectively. Curiously, a three-dimensional configuration of Borromean circles is impossible, whatever their relative sizes [4]~ This link was also known to the Norse people of Scandinavia. The symbol known as "Odin's triangle" or the "Walknot" (meaning knot of the slain) has two variants (both shown below). One is a set of Borromean triangles,
The Heisenberg Principle in Mathematics: You can know either the definitions or the theorems, but not both at the same time. Std'phane Collart Department of Mathematics Federal Institute of Technology 8092 Zurich, Switzerland
Borromean Triangles in Viking
Art Two articles in a recent issue of the Mathematical Intelligencer [1, 2] mentioned the art of John Robinson [3], in particular, his sculpture "Intuition" comprised of three interlinked triangles. They are arranged so that when taken as a whole tl~ey are inseparable but if any one component is removed then the other two fall apart. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1 (~)1995 Springer-Verlag New York 3
Symmetrical Combinations of Triangles: Postscript Peter Cromwell has noticed that the flat Borromean triad of three hollow triangles ([2], p. 29, Figure 8) was used by the ancient Scandinavians as a sign ("the knot of the slain") of their god Woden (Wagner's Wotan) (see his letter above, and [5], pp. 37, 366). H a m m a r ' s Runestone grimly depicts a sacrifice to Woden, with the sign floating above the altar. Concerning Figure 9 ([2], p. 30), Burgiel, Franzblau and Gutschera [1] cleverly use the fact that O D = O Q to prove that P and Q lie on the circumcircle of the triangle D E F . It follows that AQ = B P , y + z = t and, from Equation (3), t = cosfl + v ~ sinfl = 2cos(fl - 60~ This simple formula spectacularly supersedes (4) and yields also x =
t-1 1 2 v ~ s i n f l - 2 v ~ (c~ 1(
1-V
~1
tan
fl)
+ v~-
cosecfl)
,
the other is a trefoil knot. The two designs are clearly closely r e l a t e d - - o n e is like a superset of the other, a stellation one might say loosely. The other illustration (above) shows a detail from one of the picture-stones on Gotland, an island in the Baltic sea off of the southeast coast of Sweden. The stones were erected around the ninth century and are t h o u g h t to depict tales from the Norse myths. Some scenes s h o w fallen warriors travelling to the next world by boat or on horseback where they will join others in Odin's palace, Valh a l l a - t h e castle of the slain. The Walknot symbol is often placed near these d e p a r t e d heroes. It has also been f o u n d carved into a b e d p o s t used in a ship-burial. I w o u l d be interested to hear of any other examples of knotted or interlaced ornament.
in agreement with y + z = t. For further remarks on George O d o m , see Fowler ([3], p. 206) and Kutler ([41, pp. 19-24).
References
References
1. R. Brown, Sculptures by John Robinson at the University of Wales, Bangor, Math. Intelligencer 16 no 3 (1994) pp. 62-64 2. H. S. M. Coxeter, Symmetrical Combinations of Three or Four Hollow Triangles, Math. Intelligencer 16 no 3 (1994) pp. 25-30 3. J. Robinson, Symbolic sculptures, Edition Limitee, Carouge, Geneva (1992) 4. B. Lindstrom and H. O. Zetterstrom, Borromean circles are impossible, Amer. Math. Monthly 98 (1991), pp. 340-41
1. H. Burgiel, D. S. Franzblau and K. R. Gutschera, The mystery of the linked triangles. To appear. 2. H.S.M. Coxeter, Symmetrical combinations of three or four hollow triangles, Math. lntell. 16.3 (1994), 25-30. 3. D.H. Fowler, Analysing ancient analysis, Ancient Philosophy 7 (1987), 201-210. 4. S. S. Kutler, Brilliancies involving equilateral triangles, St. John's Review 38.3 (1989), 17-43. 5. Kveldulf Gundarsson, Teutonic Religion. Llewellyn Publishers, St. Paul, MN 55164, 1993.
Peter Cromwell Dept. of Pure Mathematics University of Liverpool P.O. Box 147 Liverpool L69 3BX England 4
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
superseding (5). Unhappily, in the approximate values given for x, p, z, the final three digits are incorrect. More accurately, x -
1 2
12
Y = ~ + -z -
4,/g 9
H. S. M. Coxeter Department of Mathematics University of Toronto Toronto, Ontm'io, M5S 1A1 Canada
~ 0.2958759, ,~ 0.8776644,
~ 1.0886621,
The Opinion column offers mathematicians the opportunity to write and neither the publisher nor the editor-in-chief endorses or accepts about any issue of interest to the international mathematical com- responsibility for them. An Opinion should be submitted to the editormunity. Disagreement and controversy are welcome. The views and in-chief, Chandler Davis. opinions expressed here, however, are exclusively those of the author
The Decimal Dysfunction Anatole Beck
After nearly two centuries, the English-speaking countries are finally adopting the metric system. The ostensible reason is that it is rational and scientific. A more probable reason is that the bulk of the rest of the world uses it and we can no longer hold out. However, in the light of science and reason, the metric system is deficient on several different counts, as I shall argue below. It is an excellent example of eighteenth-century rationalism, but in the twenty-first century it is grossly out of place. Ironically, it is an example of the dead hand of the past exerting an increasingly heavy burden on the present and the future. The basis for the metric system is the decimal system of enumeration. This in turn is founded on the biological accident of pentadactylism: we have five fingers on each hand. This fact has no more relevance than the size of the earth in the world of thought. But to the Age of Reason, it seemed anachronistic that units should be determined by the size of a king's foot, or his reach, or that fractions should be dominated by quarters and eighths, not to mention thirds, all of which was at some odds with the perceived simplicity of the decimal "Arabic" numbers. For a world which was finally coming to universal acceptance of the decimal arithmetic, a decimal system of units seemed enlightened. The basic unit of measurement was to be the meter, which was taken to be one ten-millionth of the distance from the North Pole to the eqtrator. Even if the presumed distance were correct, this was still a physically meaningless unit, as arbitrary as the foot. It did have the merit of being based on a universal measure, that of the earth we all share, rather than the dimension of some king, and an English king at that. Also, it had the merit, for Laplace and others, of correlating terrestrial distances with as-
tronomical angles, if their project of measuring angles by decimal parts of a right angle had caught on. Of course the rationalism of the eighteenth century was not to be hampered by questions of the usefulness of that particular length. Such "pragmatic" and "utilitarian" issues were more appropriate to the English, "a nation of shopkeepers," than to the olympian thinkers of the new French pantheon. The concept of natural selection had not yet made its mark on science, so it was not thought significant to consider the results of 300 (now 500) years of competition between the foot and the yard as measures of length, a competition which had left the foot in almost complete domination of the field, the yard
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1 (~ 1995 Springer-Verlag New York
5
proving convenient only for athletics, artillery, cloth, and carpets (and, today, wet concrete). Everything else was measured in feet, except for the still (in 1800) occasional use of rods, leagues, chains, versts, etc. A Darwinian rationalism would have spoken for a metric foot, perhaps a third of a meter or 30 cm., but that did not seem significant to the founders of metricism, who were determined to produce a rational system and adapt the h u m a n race to it, a la Procrustes. It seems clear that the designers of the metric system considered units to be completely arbitrary, and expected equally easy conformity to them no matter what they were. This did not prove to be the case. An example of the resistance of human beings to this Procrusteanism is to be found in Germany, where many things are measured in Pfund, no longer pounds, but instead "metric pounds" of half a kilogram. Here, the more useful unit persisted, despite its outlawry, at least in principle. People today still weigh their babies in pounds, as well as their cheeses, persisting in what the French consider an irrational archaism. It is interesting that even in the eighteenth century, there was good reason for enthroning a more significant physical unit than the size of the Earth. Assuming that one were wedded to decimalism (as the thinkers of the age universally were), one might have based the unit of length on the force of terrestrial gravity. The meter might have been chosen slightly shorter, so that the acceleration due to gravity would come out at 10 m/sec/sec. Much more reasonable, in the 21st century, would be a new foot
6
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
making that acceleration exactly 32 ft/sec/sec; more of that later. The easing of the transition between mass and weight represents a far more significant accomplishment than anything involving the distance from the equator to the pole. But all these considerations are as nothing compared to the simple fact that while ten is a number enshrined in vertebrate zoology, it has no special place in the world of ideas. It appears essentially not at all in mathematics, where the natural system of numeration is binary. Logarithms to the base ten follow the system of notation, and nothing else. Logarithms to the base two appear with great frequency and naturalness, and also the natural log of 2 itself is very significant. (An elementary example is the alternating harmonic series: 1 - 1/2 + 1/3 1/4 + . . . . ln(2).) By contrast, ten is a number of no mathematical consequence. Not only is it not prime, it is not even a power of a prime, so that it does not figure in algebra. Ten is unimportant in set theory, the very basis of counting. One might blasphemously take the importance of 2 in mathematics as a sign that God does His arithmetic in binary. One significant corollary of the dependence of set theory on the binary system is its incorporation into our computers. These machines condescend to our foolishness by allowing us to communicate with them in decimal, but then convert to their own system of notation. This should have militated against the conversion of British currency to decimal: all important arithmetic is now done by computer, which is as good at translating
into pounds, shillings, and pence as it is in converting from its own numeration into ours. Thus, it could as easily have calculated (for example) 13.5% of 176 pounds, 13 shillings, 3 pence as 12% ofs Still, the ease with which the computer converts integers between the two systems falls apart somewhat when we recall that 1.1 is not an exact number in the computer, but must be represented there by an approximation, as 1/3 is in decimal. There are, to be sure, integer systems which can deal with 1.1 as 11/10, but then those systems can also handle 1/3, so that one of the presumed benefits of the decimal system also vanishes. At bottom, mathematicians know that binary arithmetic has intrinsic validity, whereas decimal is fundamentally arbitrary. One symptom of this disparity is the new equivocation in such prefixes as kilo- and mega-. A kilogram is still 1000 grams, but a kilobyte is 1024 bytes. Similarly for mega-. We must now keep track of when the prefix means one and when the other, as we once had to know the difference between avoirdupois ounces and troy ounces. It seems clear that as time goes by, the binary usage will increasingly invade and overpower the decimal, and eventually a wrenching reform will move the world to a more binaricized system. I say binaricized rather than binary because the binary system is inconvenient precisely in the area which must have spoken most forcefully to the sages of the eighteenth century: hand calculation. Binary numbers are too long to read conveniently and too confusing to the eye. The clear compromise is a crypto-binary system, such as oc"tal or hexidecimal. I favor the former because it seems to me better adapted to hand calculation. The numbers are only about 10% longer than those in decimal, while the multiplication table (excluding the superfluous multiplications by 1 and 0) is just over half as big. (A similar comment applies to the addition table.) Octal hand calculation is actually easier than decimal, and at very small cost in the inflation of number size. As I said above, I believe that the human race will eventually yield to the more m o d e m rationality and move to a system which is binary at root. In doing so, we might reclaim the foot as a unit, probably correcting it to make the acceleration due to gravity on Earth exactly 32 ft/sec/sec. A convenient new pound might well "be 1/64 the weight of a cubic foot of water, that pound to be made up of 16 fluid ounces. If a new inch were 1/8 of a foot, then a cubic inch of water would weigh 1/8 of a pound, or 2 new ounces. 16 new ounces might be called a pint, with 8 of these making a gallon. Let's wake up and recognize that we have been throwing away a system tailored to'human dimensions in pursuit of a variety of rationalism which is two centuries out of date. In time, we will have to undo this folly.
Department of Mathematics University of Wisconsin Madison, W153706 USA THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
7
Billiards Inside a Cusp Jonathan L. King
Introduction
Here is an anecdote a b o u t how unexciting h o m e w o r k problems led a student studying calculus to a Good Question, and to the mathematics it engendered. The standard Calculus Sequence spends quite a bit of time on logarithms. Yet it is no hyperbole that the curve y = 1 / x - - a n d the standard drill questions concerning its p r o p e r t i e s - - l e a v e m a n y students comatose. One such was David F e l d m a n - - then an u n d e r g r a d u a t e at B e r k e l e y - - w h o , having complained to his dad (the mathematician Jacob Feldman) that all the h o m e w o r k was dull, dull, dull, was challenged in return to invent an interesting problem. David came u p with this:
Figure 1. For what initial position and direction will a cueball escape to infinity?
Can a mathematical cue ball (a point) fired into the symmetric funnel between y = +l/x and y = -1/x escape in any direction other than flat out? As s h o w n in Figure 1, the cue ball ricochets off the two "cushions" so that the angle of incidence always equals the angle of reflection. "Escape" means that the xcoordinate of the cue ball increases monotonically to +cx~. Evidently, a cue ball shot along the x-axis escapes. But can any cue ball which actually hits the cushions avoid being t u r n e d around eventually? Not long after it was posed, this problem was solved b y Benjamin Weiss. Unaware of its origin, or that it had been solved, or that it would eventually connect with what became m y field of study, I was intrigued by this problem w h e n Paul Shields posed it to m e during m y graduate studies. Because the problem will lead to a h a r d e r question and to the tool which is the theme of this article, I'm going to forthwith present the bare-hands solution I found at the t i m e - - so if y o u w a n t to first think about it further, read no farther. ... 8 THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1 (~)1995 Springer-Verlag New York
%%%%%
y = f(x)
(Xl' Yl
Xl
) = 1'1
I x2
x3
Figure 2. Let P~ = (xn, y~) be the coordinates of the nth reflection of the orbit of cuebalI v off the graph of f . Let a,~ be the angle that the (tangent to the) curve at P . makes w i t h the horizontal, that is, an = arctan(lf'(x~)]). After Pn, the trajectory hits the "floor," from which the cueball rises at angle 0z. A F i r s t S h o t at B i l l i a r d s i n a C u s p
Computing the slope Sn9 We have used the fact that, w h e n the cueball hits the u p p e r cushion f , the angles of incidence and reflection are equal. But s o m e w h e r e our argument had better use that these angles are equal w h e n the ball bounces u p off the floor! Here it is:
Since the problem arose in a calculus class, to get the ball rolling let's see w h a t information can be gleaned b y methods taught in an introductory calculus course 9 The argument below has become a nice capstone to the section on "Convergence tests for infinite series" in m y tan(012) -- y12 + Yn+l. o w n classes9 Xn+ 1 -- X n Because the funnel is symmetric, we can without loss , o f generality reflect the trajectory over the x-axis and, The upshot is that thus, consider the cueball to be bouncing off "cushions" tan(0n) > tan(01) y = f ( x ) = 1 / x and the x-axis. So as to focus attention Xn+ 1 -- X n on the horizontal cusp at x = +oc, let us do a w a y with Y12+ "Yn+l -- Yn q- Yn+l the vertical cusp at x = 0 b y altering the u p p e r cushion f near the origin so that it is bounded 9 This does not affect As a consequence, w h e t h e r a cueball can escape, since we keep f ( x ) = 1 / x Yn -- Y12+1 > tan(01) y12 -- Y12+1 for all large x. 8i 2 -Xnq-1 -- Xn Yn q- Yn+l Suppose, for the sake of contradiction, that in Figure 2 the cueball v bounces so that its x-coordinate n e v e r decreases: thus 0 < 012 < 7r/2 for all n, where 0n is the angle So our s u m m a t i o n condition mutates one last time, to after the nth reflection off the floor. Elementary geome- become try shows that 012 = 0n-1 + 2a12, and so the situation in Yn+l < oo, ~Yn with y12 "N O. (lc) Figure 2 implies this s u m m a t i o n condition: n=l Yn ~ Y n + l ~ a 1 2 < oc. (la) But this cannot b e - - s u c h a sum as this last one must always be infinite. Its M t h tail is Consequently Ozn --+ 0, which f o r c e s - - s i n c e f(.) is conv e x - thats xn ---* co. Thus y12 "N 0. ~ Yn -- Yn+l Y,~ -- Y12+1 "I'M = > -- -Because t a n ( a ) / a --. 1 as a --* 0, s u m m a t i o n (la) can n=. Y +I - 1 2 = . yM be restated a s E n % l [f'(Xn)[ "~ 0(3. Letting sn be the ab1 1 solute value of the slope of the line joining point P,~ with YM q- YM (YM -- lim Y12+1)= ~P12+1, the convexity of f implies that s12 _< [f'(x~)[. Consequently, oo Because the tails TM do not go to zero, conditions (lc, b, a) were all impossible, as was Figure 2. A n y cueball shot 8n < (lb) out the cusp m u s t turn around. n=l --
n
---~ O0
THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1,1995 9
Postmortem reflection. This proof can be readily shown to a second-semester calculus class and gives a nontraditional and curious use for a series divergence test. All that is used about the u p p e r cushion y = f ( x ) of the table is that f is an eventually-convex differentiable function which is asymptotic to the x-axis. However, the a r g u m e n t is unsatisfactory from the point of view of u n d e r s t a n d i n g " w h y " the cueball had to turn around. One test of the strength of a m e t h o d of argument is w h e t h e r it can be used on related questions. Suppose we remove the convexity condition a n d allow the u p p e r cushion to h a v e wiggles. Can cueballs w a n d e r monotonically out the cusp for the table determined by, sa)~ f ( x ) = (3 + sin(v/x))/(x + 1 ) 2 7
(2a)
Although one could possibly use a series-divergence arg u m e n t to exhibit a specific cueball which fails to escape, such an approach might require real delicacy to make a substantial general assertion. Yet another natural question for which the seriesdivergence approach looks ill-adapted is this stronger sense in which cueballs might fail to escape: Do cueballs return arbitrarily close (in both position and direction) to where they started?
(2b)
By the way, a cueball which infinitely often returns arbitrarily near to its initial state is called recurrent. After developing more p o w e r f u l tools, w e will come back to recurrence later.
Philosophy.
Answering questions such as (2a) and (2b) for an individual cueball m a y be difficult. Yet, nearby cueballs have nearby trajectories-- for a while - - and so it m a y be profitable to m a k e assertions about collections of cueballs: This suggests finding a useful measure on the space of c u e b a l l s - - a measure which is p r e s e r v e d u n d e r the action of "rolling" a n d "bouncing off the cushion." It turns out (this is well k n o w n to those w h o s t u d y dynamical systems but is not a commonplace a m o n g mathematicians in general) that the "billiard flow" on a n y billiard table has a natural invariant volume. The theme of this article is the tool of an invariant measure h i d d e n inside a problem which, on the surface, has no mention of measures. Along the way, w e will encounter a few elementary but useful tools from dynamical systems.
Anatomy.
In the next section, I define the billiard flow and give a pictorial p r o o f that billiard measure, which is a type of volume, is indeed invariant u n d e r the flow. Using this measure, I then present Weiss's solution to Feldman's problem w h e n the cusp has finite area, and give an almost-everywhere solution to questions (2a) and (2b). 10 THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1, 1995
In order to handle cusps with infinite area, it is advantageous to view the billiard measure differently, and for that reason the following section introduces the notion of the cross-sectional measure induced b y the billiard flow. The article culminates b y using this induced measure, a type of area, to prove that on a "pinched" table, even a table of infinite area, almost e v e r y cueball rolls recurrently. This result, illustrated in Figure 5, appears to be new. The A p p e n d i x shows connections to ergodic theory, and ends with an open problem.
History. Originally, Feldman's Billiard Problem was part of a longer article with the same theme of h i d d e n invariant measures. The other problems have been split off into a c o m p a n i o n paper, "Three Problems in Search of a Measure" [1], which applies the tool of invariant measures to Poncelet's Theorem, Tarski's Plank Problem, and Gelfand's Question. The A p p e n d i x of the present article describes a connection, in the case of an elliptical billiard table, b e t w e e n the induced measure defined here and the "Poncelet measure" of [1]. Idiosyncrasy. Use a := b to m e a n a is defined to be b. The symbol tA~176 k = l Bk means the u n i o n of sets Bk which h a p p e n to be disjoint. For a measure of "area" or "volume," a nullset will be a set which has zero area or volume. W h e n a statement "holds almost everywhere" (a.e.), this means that it holds except for a nullset of points. Reflection problems such as David Feldman's are called "billiard problems"; some curve or collection of curves forms the boundary, the cushions, and the closed 2-dimensional region lVthat they b o u n d is the billiard table. A mathematical cueball v = (v; 0} willbe a point v E F on the table together with a direction 0. If v - - sometimes called the "foot-point" of v - - i s on the b o u n d a r y of the table, then 0 is restricted to the semicircle of angles pointing into the table. All our spaces are metric spaces. A measure-space (f~, #) means that # is a Borel measure on space f~; all sets and functions are tacitly Borel measurable. A transformation T : f~ ~ f~ is a measurable map; we think of T'~(w) := T(T(.~..T(w) .. .)) as the location of w at time n. A measure # is T-invariant, or T preserves #, if -#(T-1S) = #(S) for each set S. After a cueball v has rolled for t seconds, let (I)t(v) denote the resulting cueball. This m a p p i n g r is called a "flow" and satisfies that if one flows for s seconds followed b y t seconds, the same result is obtained b y flowing (t + s) seconds. That is, a f l o w - - w h i c h is a continuous-time analogue of a t r a n s f o r m a t i o n - - o n a space f~ is a measurable m a p 9 : 1~ x f~ --, fl satisfying
such that each ~t is a transformation of f~ and G~ is the identity. Saying the flow is measure-preserving means that each ~t is a #-preserving transformation. T h e Billiard F l o w
Superficial question: What is the simplest possible billiard table? Superficial answer: One with no cushions. We first consider this minimal model of billiards. Here the table F is the entire plane R x E. Interpret IK = [0, 270 as the circle of directions (angles) 0, equipped with arclength measure dO. The space of cueballs P:=FxK is, thus, 3-dimensional and has a natural product measure vol := area x arclength, which simply measures 3-dimensional volume. Given an arbitrary set S of cueballs, its cross section in direction Ois
tsJ0:= {v r I and so vol(S) = ~
area (LS]0) dO
by Fubini's theorem. Cueball space P also has a natural topology. Letting G denote the direction of cueball v, a metric on P is
where the right-hand side uses the metrics on the plane and the "circle of directions," respectively.
Billiard flow ~. To write a formula for ~t(v), the location of cueball v after it has "rolled for t seconds" at unit speed, interpret for a m o m e n t O as the unit vector in direction 0. Then the billiard flow on the plane is the continuous m a p :=
+
tG ;
vOI(4Pt(S)) = ~ area([~tsj o) dO = ~ area(kSJ0 + tO) dO (3)
Note also: The set of cueballs which ever flow through any particular point in the table is 2-dimensional; hence it has zero volume.
Suppose the cushion of a billiard table is piecewise continuously differentiable. Then volume measure is invariant under the billiard flow. Sketch of proof of Billiard Lemma. It suffices to check that v o l ( ~ - t C ) =vol(C), where the cushion is the graph of a function f : [0, 1] ---* 1t~which is continuously differentiable, t is some fixed time, and C is a set of cueballs each of which hits f exactly once as time goes from 0 to - t . Actually, we need but verify this inequality: vol(q~-tC) _< vol(C).
Since area is translation-invariant, the flow leaves volu m e invariant:
= f ~ area([SJ 0) dO = vol(S).
The Billiard Flow Leaves vol(.) Invariant. In the case that OP consists of a single straight line, th~ argument of (3) still applies, as reflection of the plane does not change area. Together with (4), this shows that w h e n the table's cushion is a polygon, the billiard measure is ~-invariant. The lemma we shall need is that for any cushion, volume is flow-invariant. BILLIARD LEMMA.
dist(v, w) := dist(v, w) + dist(G, Ow),
9
Billiards Tables with Cushions. Moving to a less minimal case, we n o w glance at tables where reflection is possible. Suppose our table I~ c ~ x R is the closure of an open set whose b o u n d a r y OF is a nice continuously differentiable curve. W h e n a cueball hits this cushion, it keeps its tangential component of velocity but reverses its normal component. So cueball space P essentially consists of F x IK with an identification of cueball Vl with v2 if they have the same footpoint v E OF, the same tangential component of velocity, and opposite normal component. In light of (4), one can freely permit the cushion to have finitely m a n y "corners" (e.g., the origin, in Fig. 2) simply by deleting from P the nullset of cueballs which ever roll into a corner. Thus, the b o u n d a r y 0P need only be piecewise continuously differentiable. Because of the presence of corners, for a fixed t the "space m a p " v H ~ t v can jump discontinuously as the trajectory of v is moved across a corner. On the other hand, the "time map" t H ~ t v is always cogtinuous.
(4)
(5)
For then, analogous reasoning gives the same inequality with - t replaced by t and C replaced by ~ - t C . Furthermore, we m a y assume that C is a cube, because the cubes generate the Borel field on P. A "cube" is of the form C = C x I, where I c IK is an interval of directions and C is a square in the plane. We must, therefore, prove that vol(S) G vol(C),
w h e n e v e r C = ~t(s) is a cube
(5')
as in Figure 3. Suppose g : [0, 1] ~ E is a piecewise linear approximation of f , and let ~ denote the transformation of flowing for time t but bouncing off the graph of g rather than f. Because f is continuously differentiable, given e we m a y take g uniformly close to f in both position and slope so THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
11
Figure 3. A small set S of cueballs flows for time t. It bounces off the cushion, f , exactly once to form a small cube C = C x I, w h o s e interval ! of directions points to the southeast. The polygonal approximation 9 is sufficiently close to f that every cue ball of S hits 9 exactly once. The solid line s h o w s the true trajectory of v. The dotted segment s h o w s the altered trajectory of v w h e n it bounces off of g rather than f.
as to arrange, for a n y v E S, that d i s t ( ~ t ( v ) , ~ t ( v ) ) _< c. (The nullset of cueballs v which hit a vertex of g is immaterial.) This implies that ~ (S) is a subset of the set Ball~ := Ball~ (C) of cueballs which are within distance of s o m e cueball in C. But billiard m e a s u r e is p r e s e r v e d w h e n bouncing off the p o l y g o n a l cushion g a n d so vol(S) = v o l ( ~ (S)) < vol(Ball~). A n d the v o l u m e of Ballr tends to the v o l u m e of the cube Case'N0. A Second
S h o t at B i l l i a r d s
The series-divergence p r o o f of the Introduction s h o w e d that if the (piecewise smooth) u p p e r cushion d : [0, oo) --* ~ + of Fig. 2 is eventually convex, then the set of cueballs which escape is empty. For a m o r e general d, this escape set E = E(P), the set of cueballs e such that lim inf x - c o o r d ( ~ t e ) = + ~ , t---+Oo
m a y not be e m p t y but may, nonetheless, be small in another sense. A F i n i t e - A r e a C u s p H a s a N u l l E s c a p e Set: T h e S q u e e z e
Play. Replacing the convexity of d with a finite area requirement, fl ~176 f ( x ) dx < 0% allows the w e a k e r conclusion that E is a nullset. "A gallon of w a t e r w o n ' t fit inside a pint-sized cusp" is the proof: Pick :Co sufficiently large that area(S) is pintsized, w h e r e S consists of those points (z, y) E P with x > x0. Indeed, area(S) is to be taken so small that vol(S x K) = a r e a ( S ) . 27r < vol(E). 12
THE MATHEMATICAL I/qTELLIGENCER VOL. 17, NO. 1,1995
But this contradicts vol(Ot(E) N (S x IK)) ---+vol(E)
a s t ---+~ ,
which follows from the definition of the escape set. So no such x0 exists and, thus, vol(E) -- O. A S e c o n d P r o o f That vol(E) Is Zero: R e c u r r e n c e . For a continuous flow 9 on a metric space fL a point w is (topologically) recurrent if ~t, (w) s , w for a sequence of times h --* cx~. Showing that almost-every cueball is recurrent w o u l d emphatically p r o v e that the escape set is null. We will not, however, be abIe to p r o v e that E is empty, since a table of finite a r e a - - e v e n a b o u n d e d t a b l e - - n e e d not h a v e all its cueballs recurrent. 1 The k e y to showing that a.e. cueball is recurrent is to define a measure-theoretic notion of recurrence. Consider a m e a s u r e - p r e s e r v i n g flow (b on m e a s u r e space (f2~ #). A point w E S "recurs to S " if ~tw E S for arbitrarily large times t. A set S is Poincard-recurrent if a.e. w E S recurs to S. Flow ~ is conservative if e v e r y set S c f~ is Poincar6-recurrent. Define conservativity for a transformation analogously.
Consider a table bounded by a noncircular ellipse. A cuebalI hit along the major axis has a periodic orbit. On the other hand, a cueball v pointed at a focus--but with footpoint not on the major axis--has an orbit which converges to the periodic orbit along the major axis. So v is not a recurrent point. Note, though, that this example exhibits only a nullset of nonrecurrent points, as the set of cueballs pointing at a focus is but 2-dimensional.
Motivated by his study of the 3-body problem, Henri Poincar6 m a d e this simple, but tremendously useful, observation.
POINCARI~ RECURRENCE THEOREM. If 9 is a measure-preserving flow on a finite 2 measure space, then 9 is conservative. A measure-preserving transformation T on a finite measure space is likewise conservative.
they first hit the upper cushion, e hits to the right of v and, consequently, bounces off with a shallower angle than does v. Thus, after they bounce off the floor, cueball e is still further right and is shallower than v. Iterating shows that e escapes. Were such a v to exist, this reasoning would hold for the above open set of cueballs e, which perforce has positive volume. The inescapable conclusion is that E is empty.
Proof. Fix a time r > 0, and let B C S consists of those points which never recur to S after time r. Thus 9 t (B) is disjoint from B for all t _> r. Consequently, the sets /3, Or(B), 4p2r(/3), o 3 r ( B ) , . . . are mutually disjoint. Because they all have the same mass,/3 must have been a nullset. On a finite-area table, that almost every cueball is recurrent is a consequence of the following elementary fact, which is left as an exercise. LEMMA 1. Suppose f~ is a separable metric space and # is a (finite or infinite) measure. Then under any conservative measure-preserving flow/transformation on ( Yt, #), almost every point is topologically recurrent. As a consequence, because the billiard table f(x) = (3 + s i n ( v ~ ) ) / ( x + 1) 2 of question (2a) has finite area, its escape set is null. I do not k n o w whether it is empty. One can certainly manufacture a finite-area nonconvex upper cushion f which coaxes one particular cueball v monotonically out to infinity; d r a w the desired orbit of v first, then draw f to match the desired slope at the reflection points of the orbit. With a bit of extra effort, one can even arrange for f to have negative slope everywhere.
Weiss's Proof of the Empty Escape Set. Sometimes an "everywhere" rabbit can be pulled out of an "almost everywhere" hat. A case in point is the neat proof by m y friend Benjamin Weiss that under an eventually convex f of finite area the escape set is, indeed, empty.
Poincar6 Section of a Flow Imagine a large tube submerged horizontally in a river, through which water flows in some complicated way. Place a wire-mesh "surface" across the upstream end of the t u b e - - s a y , in the form of a hemisphere. Through each subregion of the mesh flows some number of gallons per minute, the "flux through the surface," which therefore specifies a measure on this surface. If we place a mesh also across the-downstream end, We get a fluxpreserving m a p (since water is incompressible, i.e., the flow is volume-preserving) from the upstream surface to the downstream surface simply by watching molecules of water flow from the one to the other. The above description is meant to motivate the definition below, where cross section E C F is a "surface," vol(E)= 0, which is transverse to the flow in an appropriate sense. Let 9 (~ (E) be the set of cueballs swept out as E flows for t seconds. [More generally, for any subset W c R of time, let o w ( E ) denote the union of ~tE, over all t E W.] The "flux" of E is the limiting rate that the volume of 9 (~ (E) grows. By this we mean flux(E) := lim 1_vol(~(o, t] (E)). t%o t
Showing that this limit exists in [0, ~ ] employs a type of argument which is frequently useful in dynamics: subadditivity. The function V[t] :=vol(O (0, t]E) is subadditive because
V[t] + V[s] = vol(O(~ > vol(O(~
The strategy is to show that if a single cueball v escapes, then an entire open set of cueballs escape. We m a y assume that the orbit of v = (v; 0~) is already in the convex part of the cusp, where the slope of f is always negative (as shown in Fig. 2), and that 0 < 0v < ~r/2. Consider any cueball e = (e; 0) with 0 > 0 and having a "shallower angle" than v, 0 < [0[ < [G[. Moreover, we ask that e lie "further right" than v in the sense that its footpoint, e, lie on the southeast side of the line through v in direction G. Compare the orbits of e and v: When
2A flow on an infinitemeasure space need not be conservative;witness (bt (x) := x + t on 9 with Lebesguemeasure.
(6)
vol(O(t't+dE) =
V[t + s].
N o w fix a positive t. Given any smaller time positive s, let N be the integer such that N s > t > (N - 1)s. By subadditivity, V[s] > (1/N)V[Ns]. Thus,
!V[s] >_ ~-~V[Ns] >_ -~sV[t] >_ N - ~ I ~V[t]. Sending s "N 0 along any sequence sends the associated N to infinity, and so liminfs\o(1/s)V[s] > (1/t)V[t]. Taking a s u p r e m u m over positive t shows that the limit in (6) always exists in [0, oo] and that
1 flux(E) = sup ~ vol(O (~ t] (E)).
(7)
t>o THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
13
FLUX PROPOSITION, 2. Suppose that L c F has an everywhere positive first-return function Tr Then (a) fux(.) is a measure on the subsets of L. (b) Volume-measure locally near L is the product measure of flux cross Lebesgue-measure on time: Suppose U C L is a subset whose first return is uniformly positive, that is, 7- := lim infvcu Tdu(v) is positive. Then
for any S c ~(--r,O] (U),
flux(U A ~ts) dt.
vol(S) = '0 q-
Figure 4. In a fixed relative direction p, assume that the points of E can be m o v e d a distance t without encountering the cushion. Then the shaded parallelogram s h o w s the location of the footpoints of those v in 5] of relative angle p, after they have flowed for at most t seconds.
Figure 5 illustrates h o w flux will be used to prove our main conservativity result, Theorem 3.
I n d u c i n g a T r a n s f o r m a t i o n . For an arbitrary cueball set E, its first-return function tells us when a cueball returns to E. The induced map, T E tells us where. It is defined on the subset of those v E E that actually return to ~Eat time T4~.(v),
(V)(v). (Of course, if ~. is a closed subset of F, then the domain of T~. is simply where T4~. is finite.) As an illustration, were E the union of the two meshes at the ends of our submerged tube, then T~. would be defined just on the upstream surface and m a p it in a 1-to-1 fashion to the downstream surface. As suggested by the physical situation of water flowing through a tube, the induced m a p preserves flux.
H o w to V i s u a l i z e t h e I n d u c e d M e a s u r e . An impor- (c) For an arbitrary set E of cueballs, the induced map T~. tant special case is w h e n we have a curve E which is a is measure-preserving wherever it is defined: If B is consubarc of the b o u n d a r y 0P, and E is the set of inwardtained in the range of T~., then pointing cueballs with footpoint on E. We can get an explicit integral for vol(E) by describing a cueball v on the flux(T l(a)) = flux(B). b o u n d a r y in terms of "relative angle." Write v = (v; p), where p C (-7r/2, ir/2) denotes the angle that v makes Conservativity on an Infinite Cusp relative to the inward normal of OF at v. Figure 4 shows an example in which E is a line segment. The area of the A case where the induced map T~. is everywhere defined parallelogram in Figure 4 is t cos(p) times the length of is w h e n ]g consists of all cueballs on the b o u n d a r y 0P. E. Multiplying by 1/t and then integrating over p gives A mutuality exists between this induced transformation and the flow: flux(Z) = f cos(p) dp dr, J(, ;p)EE Transformation Tot is conservative iff a2 is conservative.
where dv denotes arclength measure along OF and dp is arclength on ( - ~ / 2 , ~-/2). It is routine to check that this formula remains valid for a general arc E by approximating the arc by line segments and then sending t "N 0. This last step uses the premise that E has a first-return function T4~. : ~. --* [0, oc] which is everywhere positive, where T4~.(v) := sup{t > 01 ~(0, t)(v ) is disjoint fromS]}. It turns out that for an arbitrary set L of cueballs, the condition R t > 0 is a reasonable notion of cross section L being "transverse" to the flow. The proposition below is certainly plausible on physical grounds; in a n y case, it follows from standard approximation arguments applied to Eq. (7). 14
T H E M A T H E M A T I C A L INTELLIGENCER VOL. 17, N O . 1,1995
Even though Poincar6's recurrence theorem does not apply to this transformation - - the measure it preserves being infinite because OF has infinite l e n g t h - - nonetheless, on a finite-area table, Top inherits conservativity from the associated billiard flow ~. I conclude this article by arranging that an induced transformation return the favor by the following conservativity result, which appears to be new. THEOREM 3. The billiard flow under a pinched cusp, even one of infinite area, is conservative. "Pinched" means that the upper cushion f : [0, oo) --+ ~+ satisfies liminf f ( x ) = O, X---~ OO
Figure 5. To make them visible, sets E and B are shown t h i c k e n e d - - t h e y are actually subsets of "line" L1. The forward trajectory of a cueball v C B will never again touch G (although it might conceivably hit L1 elsewhere) and will sooner or later cross any given Lx. But if Lx is chosen to be a sufficiently small bottleneck, not all of B will be able to squeeze through.
as illustrated in Figure 5 below. As an example of a pinched cusp of infinite area, consider 1
f(x) := x[1 - sin(x)] + x +---~'
Moreover, S can be taken to lie to the left of some vertical line, sa~ left of x = 1. After deleting a nullset, we can assume that for all v c S, lim sup x-coord(~tv) = + ~ .
(9)
t~OO
Even though for this cushion the limsup of f(x) is infinite, nonetheless the theorem asserts that a cueball placed at a r a n d o m location and then hit in a r a n d o m direction will pass arbitrarily near to its starting position and direction. In contrast, it would seem difficult to show by means of the calculus technique of the Introduction that for this cushion there is even a single (nonperiodic) recurrent trajectory. S q u e e z e P l a y o n an I n f i n i t e C u s p . Intuitively, conservativity on a finite-area table came from being unable to squeeze a gallon into a pint-sized bottle. This time, our bottle has infinite volume but, being vague for a moment, it still has in some sense a pint-sized neck. Our gallon of water will not be able to squeeze through this bottleneck because - - if the gallon flows n o n r e c u r r e n t l y - - it has an intrinsic positive cross section that can never diminish.
Proof of Theorem 3. Suppose that ep is not conservative; then there is some set S of positive volume and some positive time 7- so that ~[r, o~)(S) is disjoint from S.
(8)
If not, then (by dropping to a positive-mass subset) all cueballs in S forever stay left of some line, x = 100 say, and ~ ( - ~ ' ~) (S) would be a ~-invariant set of finite volu m e - to which Poincar6 Recurrence would apply, contradicting statement (8). Consequently, the situation is as Figure 5 illustrates. For each positive number x, let Lx denote the set of cueballs with footpoints on the vertical line segment from (x, 0) up to (x, f(x)) and which point to the right, that is, their directions are between -7r/2 and 7r/2. This "line" is a 2-dimensional subset of cueball space. Because of Eq. (9), eventually ~tS will have positive volume lying to the right of L1. Thus, there exists a time to such that flux(E) is positive, where E := L1 N (I)t~ [This follows from breaking L1 into countably m a n y pieces U and applying Proposition 2b to each.] In addition, as assertion (8) is flow-invariant we m a y conclude that E A ~[r, ~) (E) is empty. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
15
Strategy. We have progressed from a nonrecurrent set S of positive volume to a nonrecurrent cross section G of positive flux. We will obtain a contradiction by examining the bad set B := {v e E I
=
of cueballs which never come back to G. Every such v, as Eq. (9) reminds us, eventually hits any particular line Lx to the right of L1. Thus, the induced m a p TBuL, is defined on all of B and maps it into Lx. The flow-invariance of flux n o w gives the key inequality that flux(Lx) _> flux(B),
for allx > 1.
This will flatly contradict that f( ) was pinched against the x-axis, if w e can rule out flux(B) being zero.
The infinitely-often set. Consider the set I of cueballs which, under Tr., return to E infinitely often,
I := E - (B U Tffl (B) U T~2(B) U T~3(B) U . . . ) . Of necessity, Tr. maps I into I and so 74i is finite on all of I. But I C E and, since E r~ ~[~, oo) (E) is empty, we see that T4I must, in fact, be everywhere less than 7-. Consequently, no v E I could satisfy Eq. (9). So I is e m p t y and, in consequence,
= B U Tffl(B) U T~2(B) U T:~3(B) U . . . . Because flux(E) > 0, some T~ n (B) has positive flux. And (2c) implies flux(B) _> flux(T~n(B)).
Appendix The observation that a Poincar6 section provides a fast proof of conservativity of the billiard flow in a convex oo-measure cusp arose in a discussion with m y colleague Albert Fathi. That argument led naturally to the generalization for flows on an arbitrary pinched cusp. Although it was illustrated here with a billiard flow, the conservativity result holds mutatis mutandis for a n y measure~ preserving continuous flow whose induced measure is pinched. The illustrations in this article were d r a w n with the excellent computer facilities at the Mathematical Sciences Research Institute, which I thank for its hospitality. In the setting of L e m m a 1, under a continuous flow/transformation the set of recurrent points w in addition to being a full-measure s e t - - must be residual (contain a dense G6 set), once one adds the natural assumption that # gives positive measure to every n o n e m p t y open set. In contrast, if there is no such conservative invariant measure #, then the set of recurrent points need not be residual. Nonetheless, Birkhoff established that there is at least one recurrent point under any continuous flow or transformation on a compact space. The transformation x H x + 1 on the topological circle I~ U {oo} shows that no more can be guaranteed. 16
THE MATHEMATICAL INTELLIGENCER VOL, 17, NO. 1, 1995
An unexplained coincidence occurs for flux measure in the special case where our billiard table F is b o u n d e d by an elliptical cushion C := OF. It turns out that the set C of inward pointing cueballs breaks u p into Tc-invariant subsets; one for each ellipse E which is inside of, and has the same foci as, C. The invariant set consists of those cueballs v E C whose flow trajectory will pass tangent to E before it again hits C. This invariant decomposition of C implies that flux(.), on C, breaks up into measures parametrized by confocal ellipses E. When suitably normalized, each of these measures turns out to be the "Poncelet CE-measure" of [1], which arises from what appears to be an entirely unrelated construction. Billiard flows are a kind of geodesic flow on surfaces of only zero and infinite curvature. A stronger result (see, for example, Ref. 2) is known for the geodesic flow on the surface of revolution (around the x-axis) generated by a differentiable f : [0, c~) --* ~+. If the surface is "pinched", l i m i n f ~ f ( x ) = 0, then every geodesic orbit is bounded - - except for the obvious ones which flow directly out the cusp. An open and probably difficult research question is suggested by Sullivan's result [3] on the geodesic flow on a cusp of constant negative curvature. Letting dist(v) denote the distance of the footpoint of v to some fixed point on the surface, Sullivan gives an explicit speed function D(t) such that dist(~tv) limsupt_,oo D(t) - 1
fora.e.v.
Paul Shields raised the tantalizing question of characterizing the finite-area cuspidal billiard tables which have such a speed function.
Acknowledgments This work was partially supported by NSF grant DMS9112595.
References 1. J. L. King, Three problems in search of a measure, Amer. Math. Monthly 101, #7, (1994), 609--628. 2. V.J. Donnay, Geodesic flow on the two-sphere, Part I: Positive measure entropy, Ergodic Theory Dynam. Syst. 8 (1988), 531-553. 3. D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math. 149 (1982), 215-237.
Department of Mathematics University of Florida Gainesville, FL 32611-2082, USA e-maih
[email protected]
Nineteen Problems on Elementary Geometry Armando Machado
Several years ago s o m e b o d y posed a p r o b l e m on elem e n t a r y g e o m e t r y that s e e m e d quite innocent b u t that resisted the m o r e obvious approaches. It m u s t be rather well k n o w n as it a p p e a r s repeatedly in m a t h e m a t i c a l circles. In Fig, 1, w e h a v e an isosceles triangle: ,~ = 20 ~ a = 60 ~ a n d fl = 50 ~ are given, a n d we m u s t look for the values of "y and 6. After the m o r e obvious calculations, w e obtain m o s t of the angles missing in the figure b u t not the ones w e want; all w e conclude is that "~+ 6 = a + fl = 110 ~ At this point I felt that, a l t h o u g h the p r o b l e m w a s clearly well posed, there w a s p e r h a p s no solution b y the m e t h o d s of e l e m e n t a r y geometry. I then used s o m e t r i g o n o m e t r y a n d a pocket calculator to d e t e r m i n e "), and 6: To m y as-
Figure I
tonishment, the values w e r e quite nice, namely, ~ = 80 ~ a n d 6 = 30 ~ With such values there should exist an elem e n t a r y solution! I r e m e m b e r that I used the s a m e )~ a n d a second pair of values for a and fl (I d o n ' t k n o w which a n y more) a n d that I again obtained integer values for "y a n d 6. I w a s beginning to believe that this w a s a general p h e n o m e n o n , but a third trial told m e that it w a s not so. For example, with & = 20 ~ a = 20 ~ a n d fl = 70 ~ w e obtain -~ = 2~ and 6 = 87~ . . . . I tried all the multiples of 10 ~ for ~, a, a n d r , ignoring trivial cases, like those w h e r e a = r , as well as those that w e r e s y m m e t r i c to previous ones; the o n l y data that g a v e nice values for "r a n d 6 were those in Table 1. (From n o w on, I will omit the degree symbol.) THE MATHEMATICALINTELLIGENCER VOL. 17, NO. 1 (~)1995 Springer-Verlag New York 1 7
Table I
Table 2
)~
a
fl
7
6
Problem
)~
a
fl
7
6
20 20 20 20 20 20
50 50 60 60 70 70
20 40 30 50 50 60
60 60 80 80 110 110
10 30 10 30 10 20
1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B 7A 7B 8A 8B 9A 9B 10A 10B 11A 11B 12 13A 13B 14A 14B 15A 15B 16A 16B 17A 17B 18A 18B 19A 19B 20A 20B
60/7 60/7 12 12 12 12 12 12 12 12 12 12 12 12 20 20 20 20 20 20 20 20 36 45 45 360/7 360/7 360/7 360/7 72 72 72 72 72 72 72 72 120 120
390/7 390/7 42 42 48 48 57 57 66 66 69 69 72 72 50 50 60 60 65 65 70 70 54 45 45 240/7 240/7 345/7 345/7 39 39 42 42 48 48 51 51 24 24
150/7 330/7 18 30 12 42 33 42 42 54 21 66 42 66 20 40 30 50 25 60 50 60 36 15 37.5 120/7 150/7 150/7 285/7 21 27 24 30 24 42 39 42 12 18
480/7 480/7 48 48 54 54 75 75 96 96 87 87 108 108 60 60 80 80 85 85 110 110 72 52.5 52.5 270/7 270/7 435/7 435/7 48 48 54 54 66 66 81 81 30 30
60/7 240/7 12 24 6 36 15 24 12 24 3 48 6 30 10 30 10 30 5 40 10 20 18 7.5 30 90/7 120/7 60/7 195/7 12 18 12 18 6 24 9 12 6 12
The natural conjecture was that in all these cases there should exist an elementary solution to the problem, that is to say, one that does not involve trigonometric formulas. Indeed, this was verified by one of m y colleagues, Margarita Ramalho. The interesting p h e n o m e n o n was that for each case one had to present a different proof, a situation not v e r y usual in mathematics. For example, the first two cases are quite trivial, although different from one another, and the fourth, the original one, can be solved b y superposing its figure with that of the first case, mirrored in the vertical axis, and remarking that one equilateral and several isosceles triangles s h o w up. Recently, I was playing around with the mathematical software M a t h e m a t i c a w h e n s o m e b o d y raised the same problem. I decided to use this software to look for other initial data that could lead to elementary solutions. I asked M a t h e m a t i c a to try every integer value of & and every integer or half-integer value of a and fl, choosing the cases where the corresponding values of "y and 6 were integers or half-integers. This experiment led to m a n y more candidates for an elementary solution. Following an idea of David Gale, other candidates showed up, involving 7 as a divisor of the right angle. This led eventually to the Table 2. To be strict, not all these cases are different: Cases 3A, 8A, and 12 have all the same solution and are special cases of a series w h e r e 0 < ,k < 60 is arbitrary, a = 45 + ),/4, fl = ~, "~ = 45 + 3)~/4, and 6 = )~/2. In the same way, cases 3B, 8B, and 12 admit the same solution and are special cases of a series where 0 < & < 60 is arbitrary, a = 45 + ,V4, fl = 45 - )~/4, 3, = 45 + 3)~/4, and 6 = 45 - 3&/4. Table 2 thus lists 36 problems with possibly different solutions. The n u m b e r of different problems can be further reduced. From Table 2 a kind of duality is v e r y easily discovered: To each problem ()~, a, fl, % 6), we can associate a dual ()~', a', fl', "~', 6'), with )~' = ,k, a ' = a , fl' = a - 6, " / = % and 6' = a - fl (problem 12 is self-dual). We, thus, have 18 problems once we prove this duality, this proof being the nineteenth problem referred to in the title; it can be solved by superposing the figures corresponding to (;~, a, fl, % 6) and ()~', a ~, fl', "y~,6~), after mirroring one of them in the vertical axis, and then applying a special case of Pappus's theorem from projective geometry. I must confess that I did not solve each one of these new cases b y elementary 18
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
methods (in fact, the answer given b y the computer is not a true solution, as will be evident below). This is perhaps a good activity for dead periods in academic life, such as boring meetings of the academic staff. This is also a useful illustration of the care that one must take with computer applications in mathematics. I was using the software M a t h e m a t i c a with its default precision, which gives about six correct digits, and m y p r o g r a m tested whether a n u m b e r was an integer or halfinteger or not by a m e t h o d that a m o u n t s to looking at the first five decimal digits of its double. Beside the cases in Table 2, the computer also p r o p o s e d the ones in Table 3.
lir los j)
The Gelfand Outreach Program in M a t h e m a t i c s
Table 3 A 5 5 8 8 23 23 35 39 39 41 59 59 61 61 67 67 68 68 68 77 78 97 97 129
a
fl
7
74.5 74.5 33.5 33.5 46 46 33.5 33.5 33.5 67 33.5 33.5 32.5 32.5 32 32 46 46 55.5 49.5 44 23 23 21.5
49 68 5.5 29.5 15.5 37.5 18 11.5 25 13 4 31 7.5 27.5 4 29.5 28.5 33.5 41.5 14 25.5 1.5 22 7
117 117 35 35 53 53 37 36.5 36.5 79.5 35 35 35 35 33.5 33.5 62 62 95.5 62.5 60.5 23.5 23.5 26
he need for improved mathematics education
6.5 25.5 4 28 8.5 30.5 14.5 8.5 22 0.500003 2.5 29.5 5 25 2.5 28 12.5 17.5 1.5 0.999998 9 0.999998 21.5 2.5
There are several strange things in Table 3: First, there are three values for 6 that are almost integers or halfintegers but are not exactly so; second, six dual problems are missing. This could be caused by some roundoff errors, so I used the ability of Mathematica to work with an arbitrary number of digits and tested each of the values in Tables 2 and 3 with 50-digit approximation. Every value in Table 2 remained correct, but all the values in Table 3, as well as their missing duals, appeared only as approximate integers or half-integers, although with a very good approximation. For example, the first value for ~ in Table 3 becomes 6.5000016063958834535205203604298488368773 . . . . At that moment I was astonished by what seemed an incredible coincidence: Several results that were halfintegers within five decimals but were not real halfintegers. In fact there was no occasion for astonishment: The computer had tried about 950,000 triples, many more than the number of groups of five digits.
CMAF da Universidadede Lisboa Av. Prof. Gama Pinto 2 1699 Lisboa Cedex Portugal
JJl-
T at the high school and college levels has
never been more apparent t han in the 1990's. As early as the 1960's, I.M. Gelfand and his colleagues in the USSR thought hard about this same question and developed a style for presenting basic mathematics in a dear and shnple form that engaged the curiosity and intellectual interest of thousands of high school andcollege students. These same ideas, this development, are available in the following books to any student who is willing to read, to be stimulated, and to learn.
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"This book is about Algebra. This is a very old science and its gems lost their charm foYus through everyday use. We have tried in this book to refresh them for you." -From the Introduction 1993 153 pp. Softcover $18.50 ISBN 0-8176-3677-3 1993 153 pp. Hardcover $24.50 ISBN 0-8176-3737-0
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FOR ORDERS AND INFORMATION 1 800 777-4643 or write to Birkh/iuser, 675 Massachusetts Ave, Cambridge, MA 02139.
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THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
19
David Gale*
For the general philosophy of this section see Vol. 13, no. 1 (1991). Contributors to this column who wish an acknowledgment of their contributions should enclose a self-addressed postcard.
Triangles and Computers
The Dance of the Simson Lines
Given a triangle A, let S be its circumscribed circle. From any fourth point P on S drop perpendiculars to the three sides of/X. Introduction
An earlier column [vol. 14, no. 2 (1992)] took up the subject of computer-aided discoveries in geometry and, in particular, some work of clark Kimberling on "centers" of triangles, these being points like the centroid, circumcenter, orthocenter, and so on. Kimberling defined 91 such points and found by numerical explorations that there were (or I should say, appeared to be) an enormous number of collinearities among these points. These empirical results could then be proved, again using computers, but this time using symbolic rather than numerical computation: By expressing the given centers by "trilinear coordinates" as functions of the side lengths, a, b, and c, it became a matter of showing that the appropriate determinant in these symbols vanished, a task ideally suited to programs like Mathematica or Maple. A third possible use of computer technology, one which would seem especially natural for geometric problems, is, of course, computer graphics. The first of the three examples to be presented here will show how such computer-generated pictures have led to new and quite striking results in what might be called classical Euclidean plane geometry. By contrast, our second example, although perhaps even more elementary, deals with a question quite unlike any that are taken up in any treatise on geometry of which I am aware. In the third example, pencil and straightedge replace the computer at the experimental stage; but the "punch line" is once again a result of Kimberling's numerical experiments.
* Column editor's address: Department of Mathematics, Universityof California, Berkeley,CA 94720USA. 20
THEOREM: The feet of the three perpendiculars are collinear. This locus is called the Simson line of/~ with respect to P (see Fig. 1). [According to N.A. Court, College Geometry (1925, revised 1952, N e w York: Barnes & Noble), Robert Simson (1687-1768) is wrongly credited with having discovered this line, which was actually discovered by William Wallace in 1799.] Among the Simson lines are the three (extended) altitudes of the triangle and the three extended sides. To get
P
Figure I
THE MATHEMATICAL INTELLIGENCERVOL. 17, NO. 1 (~)1995 Springer-Verlag New York
A
A
~
B
A
0
D B
D
Figure 2 the altitudes, place P at a vertex of/~ ABC. To get, say, side BC, place P at the antipode of A. (Draw the picture and you will see the simple proof of this.) Next, given an inscribed quadrilateral ABCD, four Simson lines are determined, that of/~ ABC with respect to D~ A BCD with respect to A, and so on, as in Figure 2. After looking at some pictures like those in Figure 2, Dennis Johnson noticed that the four Simson lines
seemed to be concurrent, and he proved that this was indeed the case. The result was known (Court, Theorem 304). Nevertheless, we will refer to this point of intersection as the d point of the given quadrilateral. No computers have been used so far, but Johnson next investigated the locus of the d point when A A B C is held fixed and the fourth point, P, moves around its circumcircle. A computer program produced pictures like those of Figure 3.
~176
i Figure 3 THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, l ~ S
21
As P runs around the original circle the d point appears to run around the nine-point circle of the triangle. The nine-point circle is easy to identify: It is the circle through the midpoints of the three sides of the triangle (the other six points are the feet of the altitudes and the midpoints of the lines connecting the vertices to the orthocenter). Further, the d point moves around the ninepoint circle in the same direction as P and with the same angular velocity. Johnson found proofs of all of these properties. Unfortunately, his write-up of the results was lost in the 1991 Oakland fire. The arguments, however, are not difficult if one knows some of the facts about the nine-point circle.
Next Question: What does the family of all Simson lines for a given triangle look like as P runs around its circumcircle? Johnson wrote a program which produced pictures of the "Simson family" for random triangles. Remarkably, all these line families looked the same, independent of the shape of the triangle. More precisely, all the line families appeared to be congruent, differing only in their position and orientation with respect to the given triangle. The family always had an envelope which looked as if it might be a "hypocycloid of three cusps," that is, the locus of a point on a circle which rolls on the inside of a circle of three times its radius. Figure 4 presents two examples: one for an acute, the other for an obtuse triangle. What is going on? The answer turned out once again to involve the ninepoint circle. When the graphics program was amended to include this circle, as shown in Figure 4, it was apparent that it was the inscribed circle to the envelope of the Sim-
Figure 4 22 THEMATHEMATICAL INTELLIGENCER VOL. 17, NO: 1, 1995
son family. Observing this, Johnson was able to work out completely the choreography of the dance. The ballerina, Mlle. Simson, may be thought of as carrying a long balance bar, like those used by tightrope walkers. This represents the Simson line. At each instant, as P moves around this circle the ballerina is at the d point and, hence, moves around the nine-point circle, say, clockwise with some uniform angular velocity, a;. At the same time, she is required to rotate her body counterclockwise with angular velocity w/2, in a sort of slow-motion whirl. The balance bar then sweeps out the Simson family. This gives the entire story. The three cusps of the envelope occur when the balance bar goes through the center of the nine-point circle. We will call such a line a cusp line. Suppose initially the bar goes through the center of the circle, and let this cusp line be a reference line. Then when the radial line from the center to the d point has moved through an angle O, the balance bar will make an angle (3/2)(9 with the radial line, hence it will pass through the center of the circle three times while the d point makes one revolution, producing the three cusp lines. The envelope turns out to be the hypocycloid obtained by rolling the nine-point circle (of radius 1/2) on the inside of a circle of radius 3/2. The story is not quite finished. Note that the Simson hypocycloid is an invariant of the original triangle. It is natural to ask then, for example, how from the given triangle one finds, say, its cusp lines. A final surprise: these lines are, in general, not ruler-and-compass constructible. Johnson's nice argument shows that if the cusp lines could be constructed, then one could trisect an arbitrary angle, which, as we know, is not possible with ruler and compass.
Configurations with Rational Angles For background, the reader who has not already done so should read the intriguing article "Nineteen problems in elementary geometry" by* Armando Machado on page 17 of this issue. In Figure 5, we reproduce the first of Machado's problems: Given the isoceles triangle with vertex angle 20 ~ and lines a and b making angles of 60 ~ and 50 ~ with the base, determine the indicated unknown angle 3'. It is recommended that the reader take a moment to try to answer the question in order to appreciate its difficulties. It is not hard to write various trigonometric equations which 3' must satisfy. Machado solved one of these numerically (using only a pocket calculator!) and was surprised to find that within the accuracy of the calculator, "7was 80 ~ Motivated by this discovery, he then did a numerical search using the software Mathematica and turned up several dozen other examples of these "rational configurations" where all of the angles were rational multiples of 7r. By lumping together certain families of solutions, Machado arrives at 19 distinct cases. Dennis Johnson made a computer search of a different sort, which will be described shortly, in which the lines a and b were allowed to meet the sides of the triangle not necessarily in the interior of the segments, and this turned up more than 250 examples of rational configurations. Some of these belonged to various infinite linear families of configurations, whereas others seemed to be isolated. Of course, no numerical finding, no matter how convincing, constitutes a proof that these configurations actually exist. Machado's 19 problems are, therefore, to find the existence proofs. As is usual in geometry, there are two approaches: synthetic and algebraic. The figure on the right in Figure 5 gives the picture for a synthetic proof for Machado's original example. Draw lines A B and BC, where A B makes angle of 20 ~ with the base. Then prove that all the marked segments are equal. From this and the known angles, the angle ~/can be obtained. In fact, Margarita Ramalho succeeded in finding synthetic proofs for the six Machado configurations with vertex angle 20 ~. The striking fact, however, was that each case required a different argument and a different set of sometimes as many as four auxiliary lines. It would, therefore, seem a hopeless task to try to find elementary synthetic proofs for all the examples that have been found, some, for example, with angles (k/n)Tr with denominators divisible by 7. What one would like is a uniform procedure: an algorithm for determining whether a rational configuration with given angles exists. As the unknown angle -), is uniquely determined by the given angles of Figure 5, it is clear that the four angles must satisfy some trigonometric equation. Johnson chose as parameters the angles shown in Figure 6. The relation among the angles can be expressed by various equations. Johnson finds, for example,
Figure 5
Figure 6 (sin 2 T) sin(B + E - A) = sin A sin B sin E. It turns out to be useful to convert to complex exponentials. Letting a = e 2iA, fl = e 2iB, T = e 2 i T , and e = e 2iE, one obtains (r--1)(r-l--1)[(~-lfle)--l] = (fl--1)(e--1)(~-1--1). (1) THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
23
N o w if the numbers A, B, E, and T are rational multiples of 7r, then c~, fl, 7-, and ~ are roots of unity. For example, in the original problem, A = (1/3)~r, B = (5/18)~r, T = (4/9)~r, and E = (1/6)7r, with lowest common denominator 18. Taking ~ to be a primitive 18th root of unity, we get c~ = ~6, fl = ~5, ~_ = ~s, and e = ~3. Substituting these in Eq. (1) gives the polynomial equation ~17 q_ ~15 _ 2~12 + 2~8 _ ~5 _ ~3 q_ ~2 _ 1 = 0. (2)
b
a
c
To show that the equation is satisfied, we check that it is divisible by ~6 _ ~3 q_ 1, the cyclotomic polynomial for the primitive 18th roots of 1. In fact, Eq. (2) factors as (~6 _ ~3 + 1)(~11 + ~9 + ~s _ ~6 _ 2~3 + ~2 _ 1). The method is clearly general and was used by Johnson to find all of his rational configurations. For each positive integer N, let A, B, T, and E take all integer values k < N. Then substitute the corresponding powers of ~ in Eq. (1) and check to see whether it is divisible by the Nth cyclotomic polynomial . Unlike the numerical search, this method also proves that the configurations exist. (Using a similar technique, Raphael Robinson also verified the existence of all of the configurations in Machado's article.) Despite the mass of data now available, the structure of the set of rational configurations remains quite mysterious. Johnson found rational configurations for all even values of N from 8 up to 36, but there are none for odd N in this range (aside from trivial cases where, e.g., c~ = fl). Are there perhaps no nontrivial rational configurations with N odd, and if so why? Also mysterious is the source of the original Machado example. Any references readers can supply regarding the history of the problem would be of interest.
Triangles Within Triangles All right, boys and girls, today we're going to do an experiment in geometry. N o w I know you all know how to write computer programs that draw triangles and perpendicular lines, but today I'm going to show you a different method. All you need is a pencil (you remember them, don't you?) and a straightedge which can draw right angles. You each have a piece of paper showing the same triangle with sides a, b, and c. N o w please follow my instructions. I want you to choose any point on your paper and draw a perpendicular to line a. Got it? From there, draw a perpendicular to line b, from there, a perpendicular to line c, and then a perpendicular back to line a, and keep on drawing perpendiculars, 'round and "round. What do you notice? That's right, Karl Friedrich, after a while you keep getting the same triangle inside the one you started with. And, yes, it seems to be a smaller version of the original triangle turned on its side. (See Fig. 7.) Please notice that all of you got the same triangle even though you chose different starting points! Why is this, do you suppose? What's that, Henri? A "contraction mapping"? Hmmm. 24 THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1,1995
Figure 7
P
/ Figure 8 All right, now we're going to try something a little different. Start again with a new starting point, but this time instead of drawing your perpendiculars from a to b to c, go the other way, first to a, then to c, then to b, so you spiral around in the opposite direction. Right you are, Sonya, this time we get a different triangle, but it's really the one we got the first time only it's standing on its head. (See Fig. 8.) These phenomena were discovered by Hidefumi Katsuura. The fact that the two limit triangles are similar to the original rotated through 4-90 ~ is clear, but Katsuura shows that they are, in fact, congruent; they have the same circumcircle and are related by symmetry with respect to the center of this circle. Further, the image of the original triangle under this symmetry contains the six vertices of the two limit triangles. All of this is illustrated in Figure 9. Up to this point computers have not entered the picture. Note, however, that the center of symmetry P of Figure 9 is an invariant of the original triangle. The natural question arises, which center is it? The centroid, the
Figure 9 orthocenter, incenter, circumcenter, nine-point center? To answer this question, I did the obvious thing and dashed off an e-mail to Clark Kimberling in Evansville, the previously mentioned world expert on centers of triangles, describing the construction. Kimberling ran a few numerical experiments and by return e-mail informed me that the point was none of the above, but its coordinates agreed with those of the symmedian or Lemoinepoint to 14 decimal places! What! You never heard of the symmedian point? The symmedian point is the conjugate point of the centroid. Conjugate point? OK, chose any point P which is not one of the vertices A, B, and C of the triangle. Reflect " lines PA, PB, and PC in the angle bisectors at A, B, and C, respectively. The lines so obtained are (theorem) concurrent, and their intersection, P~, is the conjugate of P. Knowing, or I should say suspecting, that P is the Lemoine point, one can prove analytically that it actually is, as did Clifford Gardner, or better still, one can turn again to N.A. Court, Chapter 10, "Recent Geometry of the Triangle," Section B, "Lemoine Geometry" (Emile Lemoine, 1840-1912), Theorem 593, from which the Katsuura results follow easily. The circle in the figure is known as Lemoine's second circle, and the three diameters are the Lemoineantiparallels. Katsuura, however, gives direct elementary proofs of his results, so knowledge of 19th-century mathematics is not needed.
Figure A1
Addendum: Jigsaw Paradoxes Our first section was devoted to demonstrating the power of the use of computer graphics in solving geometric problems. We will ~onclude with a demonstration of the power of the misuse of computer graphics to provide fallacious solutions. Figure A1, which has been attributed to Lewis Carroll, is a well-known mathematical hoax. It claims to prove that area is not necessarily preserved under finite decomposition.
Figure A2 THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
25
Figure A3
The reader who has not seen this before should try to find the fly in the ointment. The same general principle has been used to create other such paradoxical decompositions as Figure A2. A much more elaborate variation on this theme was invented last year by Jean Brette, who for many years has been in charge of mathematics at the Palais de la Ddcouverte, the science museum of Paris. The example, which is illustrated in Figures A3 and A4, is actually six paradoxes in one. Using subsets of the 10 pieces shown in Figure A3, one can assemble a 9 x 16 triangle in six different ways, corresponding to the six orderings of the three triangular pieces along the hypotenuse of the big triangle. Referring to Figures A3 and A4, note that the nontriangular pieces of the bottom right triangle have total area 44, those of the right-hand triangle have area 45, the next 46, the next 47, the next 48, and the lower left 49. You might try making your own set of pieces out of cardboard. Amaze your friends! In the dissection described here the trick is relatively apparent. However, Brette has found a general method for constructing these paradoxical dissections for any right triangle with integral sides, and it is thus possible to construct examples where the three triangles along the hypotenuse are so close to being similar to the big triangle that the discrepancy in shape becomes essentially imperceptible. Figure A4
26 THEMATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
The Impossible Problem Lee Sallows
Miracles we perform instantly, the impossible may take a leetle longer. (author's motto) Leafing through back issues of Scientific American recently I came across an intriguing c o n u n d r u m d u b b e d "The Impossible Problem" in Martin G a r d n e r ' s Mathematical Games d e p a r t m e n t for December 1979. Then, as now, Gardner was the leading figure in recreational mathematics, his regular column famous as a trading centre in offbeat and exotic ideas. The Impossible Problem was new to me. "This beautiful problem," wrote Gardner, "I call 'impossible' because it seems to lack sufficient information for a solution." I could only agree: one reading and I was seriously hooked. "If there is a simpler solution than the one given I should like to k n o w about it," he wrote. Here is the problem exactly as he presented it:
tory. A n d the problem l i v e d up to its name. Its solution was not only elegant, it called for some intricate thinking. Having t r i u m p h e d at last, I then reached for Mathematical Games to see how G a r d n e r ' s approach'compared. A surprise awaited me: his answer was different from mine. I was less fazed by this than y o u might think. That a second answer based on a different kind of argument might exist had already crossed m y mind. After all, the puzzle tells us a story about two people and some things they said to each other. Then w e are asked, "What are the two numbers?" However, the two numbers referred to here never actually come into the story. What
Two numbers (not necessarily different) are chosen from the range of positive integers greater than I and not greater than 20. Only the sum of the two numbers is given to mathematician S. Only the product of the two is given to mathematician P. On the telephone S says to P, "I see no way you can determine my sum." An hour later P calls him back to say, "I know your sum." Later S calls P again to report, "Now I know your product." What are the two numbers? It took me four days to crack this nut. H a l f w a y throtigh I even wrote a c o m p u t e r p r o g r a m to assist the process. This was heavy-handed, I admit, but after two days w i t h o u t a breakthrough desperation was setting in. H a d Gardner not e m p h a s i z e d its seeming impossibility, I might have t h r o w n in the towel; only his assurance there was a solution kept me going. The c o m p u t e r print-out m a d e it easier to s u r v e y relations a m o n g sums and products; it played no decisive role in licking the problem but it did ease me toward an insight that led to eventual vicTHE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1 ~) 1995 Springer-VerlagNew York 2 7
the question really boils down to is: Can you discover two numbers that consistently explain all the facts presented? Apparently Gardner could produce another pair that worked as well. Nonetheless, it seemed remarkable. To find the one solution had demanded hours of concentrated attack; the notion that an alternative existed strained credulity. I could hardly wait to read his account. Starting in, however, I soon found myself baffled by his argument. As far as I could see it just didn't add up. Try as I might, I could not go along with his logic. Suddenly I had an idea. Of course: it had to be an error. A correction was certain to be found in a subsequent column where all would be explained. I immediately began looking through succeeding issues of Mathematical Games. Sure enough, there it was in a postscript at the end of the column for March 1980: "As hundreds of readers have pointed out," I read, "the 'impossible problem' given in this department for December turned out to be literally impossible." Literally impossible? I goggled. "Because I gave an upper bound of 20 for the two selected numbers," he continued, "the solution became totally inapplicable." I thought this over and it began to make sense; this matter of the upper bound had been mentioned previously: "To simplify the problem I have given it here with an upper bound of 20 . . . . If you succeed in finding the unique solution, you will see how easily the problem can be extended by raising the upper bound. Surprisingly, if the bound is raised to 100, the answer remains the same." 100 had been its value in the original version of the problem as first passed on to him by a correspondent. Only now had he realized that it could not be lowered without incurring disaster. In changing the ceiling from 100 to 20 he had inadvertently made it impossible to eliminate certain candidate solutions, and thus made it impossible to solve the problem. Or so he thought. It was a natural assumption for o n e who believed the intended solution was unique~ I had therefore discovered something that Martin Gardner never guessed. His Impossible Problem with its lower bound of 20 is not insoluble. But it is a tough cookie, in my estimation at least. Note carefully that I refer here to the problem exactly as reproduced above and not to any supposed equivalent or variation. In particular, the above should not be confused with its progenitor, the "same" problem that Gardner had received from a correspondent, the original publication of which he was able to announce later in a second postscript. In the sequel we shall see that in reworking this problem for presentation in Mathematical Games, Gardner changed more than the upper bound, but without ever realising that in so doing he was admitting a new kind of solution. Readers who enjoy a challenge may like to try their hands at the problem before comparing notes with the solution that follows. Table 1 surveys relations among the sums and products involved, and may prove helpful. It is the print-out produced by the program mentioned. 28
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
T h e S o l u t i o n to t h e P r o b l e m
I said Gardner's changes in presentation had created a new problem. What is remarkable is that the new problem accidentally created should turn out to be as "impossible" as the original. In tackling it however we shall proceed on the usual assumption that it is a deliberately and carefully constructed puzzle. In discussing its solution I shall use valnum as shorthand for valid number, meaning any integer greater than I and not greater than 20. Let/9 stand for the product, which must lie between 2 x 2 = 4 and 20 x 20 = 400, and s for the sum, which lies between 2 + 2 = 4 and 20 + 20 = 40; x and y are the two unknown numbers. To begin with, suppose x and y were the only two valnums (not necessarily distinct) whose product is p. Call such a product unique. This may entail that x and y are both prime numbers, but need not: 8 is unique since 2 x 4 is the only product of two valnums that will produce 8, and 4 is non-prime. Now if P's product is unique he could have factored p and identified x and y immediately. But P only deduces the sum an hour after hearing S's first remark. So p must be the product of at least two distinct pairs of valnums, and S's statement must convey some information that makes it possible for P to select the correct pair from among different candidates. Yet all S says is "I see no way you can determine my sum."
A slip in the formulation of a near impossible p u z z l e m a d e it actually unsolvable. Or did it? At first sight it is hard to see any useful information conveyed by this. What can S's estimate of P's ability to determine her sum tell P that he doesn't know already? Note however that the statement is made on the telephone. It may seem that the telephone is a mere incidental feature of the problem, yet an answer that can exploit every detail is better than one that cannot. Thus, equipped with a telephone, S has had a chance to wait awhile before dialing P's number. P might have called first, but didn't. Without the telephone as a giveaway, we might not have known that it was possible for S to pause and see whether P would respond quickly first. In the meantime S could have listed each of the possible pairs of valnums whose sum is s and noted their corresponding products. The latter may include unique products, but reasoning as above, S will know that p cannot be one of these since otherwise P would have already phoned to say, "I know your sum," or said the same right after he heard that it was S on the line. Hence S's list must also contain one or more non-unique or ambiguous products, among them p. This raises a special case. For in the event that there were only one ambiguous product, S would then know that it had to be p. For example, suppose s is 7. The possible pairs of valnums that add to 7 are 2 + 5 and 3 + 4. Their corresponding products
8urea
suns
~trm
9
2,2
9
5
2,3
6
6
219
:
6
3,3
7 7
2,5 3,4
8
2
8 8
3,5
16
4,4
16
-- 2 X 8
9
9
2,? 3.6
14 18
"
9
4,~
20
1
10 10 10
p
6
r
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10 12
~
2x6
12
~
3x4
2x9 2X10
16 21 24 26
--
4x4
~
2X12
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2,8 3.? 4,6 5 5
11 11 11 11
2,9 3,8 4,? S 6
18 24 28 30
1 ----
3X6 2X12 2X14 2X15
12 12 12 12 12
2,10 3,9 4,8 5,7 6,6
20 27
-
4x5
32 35 36
m 2X16
2X18
-- 3 x 1 2
:
4X9
13 13 13 13
2,11 3,10 4,9 5,8
22 30 36 40
- 2x18 -- 2 X 1 8 - 2X20
~ 5X6 ~ 3x12 -- 4 x l O
i
6X6
13
6,7
42
m
3x14
14
2,12
24
l
3 x8
14 14 14 14 14
3,11 9 S 9 6,8 7,7
33 40 1 2x20 4 5 1 3 x15 48 m 3X16 49
15 15 15
2,13 3.12 4,11
26 36 44
1
15 15 15
5,10 6.9 ?,8
54 56
-- 3 x 1 8 -- 4 X 1 4
16 16 16
2,1 9 3,13 4,12
16 16 16 16
5,11 6.10 ?,9 8,8
17 17 17
-
-- 3 X 8
m
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4 x6
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60 63 64
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m
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2,15 3,1 9 9
30 42 52
- 3XlO -- 6 X 7
-- 5 x 6
17 17 17 17
5,12 6,11 7,10 6,9
60
-- 3 x 2 0
-
4x15
5X14 4x18
m
6X12
18 18 18 18 18 16 18 18
2,16 3.15 4,14 5.13
32 46 56 65
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l
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?.ii 8,10 9,9
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34
19 19 19 19 19 19 19
2,17 3,16 9 5,14 6,13 7,12 8,11 9,10
90
- 5x18
~ 6x15
20
2,18
36
1
I
20 20 20 20 20 20
61 64 75 84 91 96
20
3,17 4,16 5,15 6,14 7,13 8,12 9.11 10,10
21 21 21
2,19 3.18 4,17
54 68
~ 6X9
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80
1
21 21 21
6,~5 7.14
90 98
1
8,13 9,12 10,11
104 108 110
2,20
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3,19 9 5,17 6,16
5? 72 85 96
20
21
21 22 22 22 22 22
72
80
m m
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48 ?0 78
84
3,20 4:19 5,18
60 ?6 90
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102
23 23 23 23
7,16 8,15 9,14 10,13
112 120 126 130
23
11,12
132
29 24 24 24 24 24 24 24 24
9 5,19 6,18 ?,17 8,16 9,15 10,14 11,13 12,12
25 25 25 25 25 25 26 25
-- 8 X l S
1
4X15
1
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~
6x15
-
9X10
1 1
8X14 6X20 7X18
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m
80 95 108 119 128 135 140 19 144
1
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--
9x12
1
8X18
"
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5,20
100
-
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6,19 ?,18 8,17 9,16 10,15 11,14 12,13
114 126 136 144 150 18 9 156 120
26 26 26
6.20 ?,19 8.18 9,17 10,16 11,15 12,1 9 13.13
27 27
7,20 8,19
19 152
27 27 2? 27 27
9,18 I0,i? Ii,16 12,15 13,14
162 I?0 176 160 182
28 28
8,20 9,19 10,18 11,17 12,16 13,16 14,14
160
28 28 28 28 28 29 29 29 29 29 29
9,20 10,19 11,18 12,17 13.16 14,15
180 190 198 20 9 208 210
30 30
10,20
200 209
4X12 3X20 ?XlO
--
6X14
1 1
6X8
5X12
m
6xlO
88
3X12
4 x9
l
6 x6
-- 8 x 8 m
7x12
~
6x16
99
-
i~
5x20
38
26 26 26
m
--
1
133 144 15~ 160 165 168 169
I?I 180 187 192 195 196
30 30 30 30
ii,19 12,18 13,17 19 16,15
31 31 31 31 31
11,20 12,19 13,18 14,17 15,16
220 228 234 238 240
32 32 32
12,20 13,19 14,18
29 24? 252
32 32
15,17 16,16
256 286
33 33 33 33
13,20 14,19 15,18 16,17
260 266 270 272
34 34 34
14,20 15,19 16,18
280 285 288
34
17,17
289
35 35 36
15.20 16,19 i?,18
300 30 9 306
36 36
16,20 i?,19 18.18
323 324
3? 37
17,20 18,19
39 342
38
38
18,20 19,19
360 361
39
19.20
380
40
20,20
9
36
4x20 5 xI8
6x10
-- ? x 2 0
-- 9 X l 9 i
8X18
-
12x12
1
8X15
"
10X12
1
9X16
"
12X12
1
8X20
"
lOxl 9
I
9x20
1
10 x18
--
10x16
m
9X20
m 12x15
-- l O x 1 8
:
12x15
-- 6 X l O
4x8 m 5X9 -- ? x 8
m --
1
5x12
81
60
23 23 23
-- 7 x 1 6 m 6X20
26
m 4x12
28 39 48 55
66 70
105 112 117 120 121
4X6
1
--
prod~-"t.s
?,15 8,14 9:13 10,12 11,11
26
2X18
pairs
22 22 22 22 22
8xlO 9xlO
216 221 22 9 225
m
12x20
-- 1 5 X 1 6
320
6x18
1
4XlO
-- 5 x 8
1
6x12
1
8X9
~ 8X12
Table 1. The 40 possible sums and their associated products. THE
MATHEMATICAL
INTELLIGENCER
VOL. 17, NO. 1,1995
29
are 10 and 12. 10 is unique, while 12 = 2 x 6 = 3 x 4 is ambiguous. Were P's product 10 then P could work out in a flash that the two numbers are 2 and 5. But should S not hear from P fairly soon then she will reason that P must have 12. A solitary ambiguous product on S's list will always allow her to name p. The question is, though: is S able to name p at the time of her first call? The implication of what she says may or may not have been consciously intended by her, but is inescapable: No. For in not saying "I know your product," she reveals to P that she cannot yet name it, a fact subsequently confirmed by her second call: "Now I know your product." Of course, P might have concluded the same had S remained silent for long enough, but as it happens, S phones first. Until S speaks, for all P knows she could ring up at any moment to name his product. S's first call is able to resolve P's doubt. Here then is a piece of incidental information conveyed by S's initial remark, a tiny tidbit, but the key that we shall need. Granted that S might have said, "The walls are very perpendicular tonight," or almost anything else, and the tacit implication would have remained unchanged. Bear in mind, however, that any irrelevant remark would have alerted P, as it would have alerted us, that something surreptitious was afoot. As things stand, S's remark enables P to infer something he didn't know before, while by the choice of words, "I see no way you can determine my sum," we detect nothing untoward. The statement is a decoy, in that it says one thing while it means another. Put different words into S's mouth and the problem becomes more tractable at the expense of its "impossibility". Thus, despite its unpromising appearance, S's call has yielded a morsel of data for P. It is the merest crumb, but one that P might be able to use under special circumstances. For P can list the possible pairs of valnums whose product is p and note their corresponding sums. Taking each sum in turn, P can now put himself in S's shoes and table what would then be S's candidate products. Like us, P will have inferred that S's actual list must show more than one ambiguous product. Were it the case that one, and only one, of P's candidate sums gave rise to a list for S showing more than one ambiguous product then that sum would have to be s. Accordingly, our next question becomes: is there a product that could have placed P in this position? P's product lies between 4 and 400. Starting with the smallest, consider the possibilities in turn. Prime numbers and unique products can be ruled out, which disposes of 4, 5, 6, 7, 8, 9, 10, and 11. Next comes 12. If P's product is 12 then x and y can only be 3 and 4, or 2 and 6. The corresponding sums are 7 and 8. We have just looked at the case when S has 7; it results in one ambiguous product. Thus, since S's remark has shown that she cannot identify his product, P now knows that 7 is not the sum. But this would tell him that it has to be 8. Can it really be so?
30 THEMATHEMATICALINTELLIGENCERVOL.17,NO.1,1995
We can check this against the foregoing. The pairs that sum to 8 are 2 + 6, 3 + 5, and 4 + 4. Their corresponding products are 12, 15, and 16. 15 is unique. But 12 = 2 x 6 = 3 x 4 and 16 = 4 x 4 = 2 x 8 are ambiguous. 8 is thus the only one of P's two candidate sums to give rise to a list for S showing more than one ambiguous product, tt has worked exactly as predicted. We seem to have struck lucky amazingly quickly. Given 12, then once he knows that S cannot name his product, P can deduce that S's sum is 8. It takes P an hour to do it, but then the underlying import of S's remark will not have sunk in at once. Can S identify P's product when given 8? No. It might be 12, it might be 16. Given 8, all she might do is to tease P in a feminine way with her seemingly innocuous "I see no way you can determine m y sum". No way, that is, until P calls her back to say, "I know your sum". For that would give S a fresh insight. Now S is quite capable of working out the foregoing chain of argument for herself. Her discovery that a product of 12 is the only one of her two candidates, 12 and 16, that would have allowed P to name her sum is but a matter of time. At that point she phones him again to say, "Now I know your product." Everything is now explained. P has 12, S has 8, the two numbers are 2 and 6. One solution is thus 2 and 6, but is this the only pair that works? We had hardly begun checking out P's possible products; what happens beyond 12? An hour suffices to run through the remaining cases. My result can be checked by others. No further product will turn the same trick; 2 and 6 form the sole solution of its kind. However he did it, Martin Gardner has bequeathed us a classic gem.
Reconstructing the Crime I have said The Impossible Problem was the accidental fruit of changes Gardner introduced in presenting another problem. The time has come to examine this progenitor in detail. In the foregoing it has been convenient to speak of "Gardner's solution," but of course Gardner was merely reporting the known answer to that earlier problem. As his second postscript in Mathematical Games for May 1980 informs us, the earliest known appearance of the original problem is due to Hans Freudenthal, who presented it in Nieuw Archief Voor Wiskunde (Series 3, Vol. 17, 1969, p. 152). Two solutions received from readers, at root identical, were printed afterwards in the same Dutch journal (Vol. 18, 1970, pp. 102-6). What looks like an English translation of Freudenthal's problem then appeared six years later in Mathematics Magazine (Vol. 49, No. 2, March 1976, p. 96), submitted by David J. Sprows. The solution given is again the same, the one that Gardner describes. The latter publication would seem to be the most likely source tapped by Mel Stover, the Winnipeg correspondent who brought it to Gardner's attention. A comparison between this and the Mathematical Games version
of three years later reveals some interesting differences. Here is the F r e u d e n t h a l / S p r o w s problem: Let x and y be two numbers with I < x < y and x + y ~ 100. Suppose S is given the value x + y and P is given the value xy. (1) P says: "I don't know the values of x and y." (2) S replies: "I knew that you didn't know the values." (3) P responds: "Oh, then I do know the values of x and p." (4) S exclaims: "Oh, then so do I." What are the values of x and y? The close resemblance between this and The Impossible Problem is clear at a glance. The two, however, are distinct. We shall not examine the lengthy solution to this problem here, details of which can be f o u n d in the references cited. In the first place, both the n u m b e r of statements m a d e and the order of the speakers in the two dialogs differ. Assuming the above was the dialog Gardner started with, we can imagine him thinking to himself that clarity w o u l d be gained b y switching S's first statement with P ' s in order to rid the former of its retrospective stance. S's "I knew that y o u d i d n ' t k n o w the values of x and y" w o u l d then become, "I see no w a y you can determine m y sum." If P cannot determine x and y then he cannot determine their sum. But having done this, Gardner will have seen that P ' s statement (1) then becomes wholly r e d u n d a n t and can be dropped. The result is his Mathematical Games version using only three statements, which is admirably succinct. Succinct yet different, as becomes clear w h e n w e try to a p p l y the same reasoning that is used successfully in the solution to the original problem. Thus, consider the opening inference in G a r d n e r ' s o w n words: "After S said 'I see no w a y you can determine m y sum,' P quickly realized that the s u m cannot be the sum of two primes." Starting from statements (1) and (2) above, this deduction of P ' s w o u l d make sense. S's "I knew..." reveals she was aware p could not be factored into two primes before deducing the same via statement (1), a conclusion she could only have arrived at from contemplating s alone. But in Gardner's new version of the problem the same inference is really a bit silly, since it overlooks the practical point that P w o u l d have simply n a m e d the s u m first, had he been able to factor p into two primes. Hence G a r d n e r ' s tinkering has already altered the problem, quite apart from his change to the u p p e r bound. Secondly, Gardner does more than just lower that upper bound. In the above problem the b o u n d is defined differently: it is x § y that r~ust not exceed 100. Notwithstanding, G a r d n e r ' s definition seems to me the more natural, but what of his choice of 20? What happens if the u p p e r b o u n d is varied? A computer p r o g r a m I wrote that is able to scan for solutions w h e n different b o u n d s are imposed has revealed a surprising fact: the solution of 2 and 6 remains unaffected by the value, p r o v i d e d it is not
less than eight. Whatever the ceiling value b e y o n d this lower limit, 2 and 6 is always a solution, so that the very stipulation of an u p p e r b o u n d is really superfluous. On the other hand, since products that are unique for one u p p e r b o u n d m a y become ambiguous with another, and vice versa, then depending on the b o u n d in force, extra solutions can be created. In fact multiple solutions turn out to be the rule, as s h o w n in Table 2. Even lowering the b o u n d to certain values below 20 gives rise to more solutions, while b e y o n d 20, 24 yields five, 50 yields six, and 84 yields seven. What distinguishes all these extra solutions from 2 and 6, however, is their bound-dependence. Most interesting of all though, is to discover that Gardner's choice of 20 is one a m o n g only 15 u p p e r b o u n d values below 100 to result in the unique solution of 2 and 6. Ironically, 100 is another instance, so luck was on his side again. Thirdly, unlike Gardner, F r e u d e n t h a l / S p r o w s dem a n d that the two numbers, x and y, be distinct. This is no trivial point. H a d Gardner not a d d e d "not necessarily different," The r-mpossible Problem w o u l d be truly insoluble. This is because S's sum, 8, could no longer be 4 + 4, a change that completely u n d e r m i n e s our solution method. So once more, G a r d n e r ' s decision was crucial. Fourthly, in a surreal move reminiscent of Salvador Dali, Gardner introduces a telephone into the landscape. Its role is twofold, I guess. It is a w a y of indicating that S and P are unable to see each other's number, but it also tends to humanise the disembodied sounding utterances of F r e u d e n t h a l / S p r o w s ' dialog, whose version was pitched at a mathematical audience, remember. Yet oh h o w snugly the telephone fits into the reconstruction of events as achieved in our n e w solution. There is S awaiting the call that will tell her P can name her sum. Time goes b y and nothing happens. After concluding he cannot do it she decides to call him. H a d this been cast in the disembodied utterance m o d e y o u could never be sure w h e t h e r S had had an o p p o r t u n i t y to wait for P to speak first. The telephone guarantees that opportunity. G a r d n e r ' s telephone pours oil on the cogs of cognition as they grind toward a solution. So to s u m up: four main points distinguish G a r d n e r ' s formulation of the problem from that of its original: the structure of the dialog, the u p p e r b o u n d , the distinctness of x and y, and the telephone. N o t a one of these changes was necessitated; rather they are arbitrary, or the result of personal taste or whim. Overlooking for a m o m e n t what went wrong, certainly Gardner p r o d u c e d a crisper c o n u n d r u m for his readers, but that might equally have been achieved in a h u n d r e d different ways. Coincidence is too w e a k a word to describe what has happened. It is almost as if some unseen force has guided the constructor's hand. Only the delicate combination of those particular changes he w r o u g h t has conspired to produce The Impossible Problem. Vary or omit but a single detail and the problem disappears because it cannot be solved, or it has too m a n y solutions. A d d to this that the new THEMATHEMATICAL INTELLIGENCER VOL.17,NO.1,1995 31
puzzle thus created, with its devilish decoy realised in the sneaky significance of S's remark, is itself even worthier than its prototype of the name "Impossible", and the whole series of events is revealed as nothing short of miraculous.
<8 8
9 10 11 12 13 14 15 16 17 18 19 20 2Z 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 S6 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 ?2 73 ?4 ?5 ?6 77 78 ?9 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 i00
nonl 2,6
2,6 2.6 2,6 2,6 2,6 2.6 2,6 2.6 2,6 2,6 2,6 2.6 2.6 2,6 2.6 2,6 2.6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2.6 2,6 2,6 2,6 2,6 2.6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2.6 2,6 2,6 2,6 2.6 2,6 2,6 2.6 2,6 2,6 2,6 2,6 2,6 2.6 2,6 2,6 2.6 2,6 2,6 2,6 2,6 2,6 2.6 2.6 2.6 2,6 2,6 2,6 2.6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2,6 2.6 2,6 2,6 2,6 2,6 2,6 2n5 2,6 2,6 2,6 2,6
4,6
2,9 5,8 5,8 8.9 8,9 7.12 ?,12
9.10
11,16 11n16 12:22
14.18 14,18 14.18 14.24
16,18 16.18
~6.25 16.27 16,27 20.30 22.27 22,27 22.28 24.30 24,30 26.36 26,36 26,36 26,36 27t45 33,40 33,40 33,40 36,36 36,36 35.42 36,49 36.49 36,49 42.45 40.54 40,54 36,56 36,56 36,56 44,60 44.60 44,60
30.30 30,30 30,30 27,40 27.40 25.42 25.42 35.36
30.32 30,32
32,35 32,35
36,40 35,42 36,44
36.49
40,42
44,45
44.50
45,52 45,52 45.52
48.50 48,50 48.50 48,50 48.56 48,56
50,54 50,54 50,54
60.63 60.63 60.63 60,63 60.70
63,64 63,64 63,64 63,64 63.64 63,64
45,64
55,56 55,56 49.72 49,72 49,72 52,75 54.76
56,60 56,60 55,72 55.72 55,72 55,72 55,72 55,72 65,66 65,66 65,66
56,81 56,81 56.81 60.84
63,76
60,88 60,88 75.78
64,95
The Impossible Problem came about as a fluke. Can deliberate changes improve its formulation? Trying out various schemes, one thing led to another until I noticed that the principle at work in its solution can be extended to create a new problem that is even deeper. The result is quite amusing in that it presents a similar situation and demands a similar answer, while providing quite a bit less information than before. Indeed, The Superimpossible Problem, as it may aptly be called, appears so utterly incapable of solution as to suggest a joke. Need I emphasize, therefore, that the problem below certainly does provide sufficient information to reach an answer by means of straightforward reasoning without resort to hanky-panky? The solution, which will prove the point, follows immediately after, so that prospective solvers should be sure to cover it up now before reading further. Here is the problem: A wealthy patron, A, invited two mathematicians, P and S, to take part in a special contest. P and S knew they were to be opponents, but there was no prior contact between them.
44,65 48,63 48,63
60,77 60,77
18j20 18.20 18,20
A Superimpossible Problem
72,80 72.80 72,80 72,80
65.72 65,72 65.72 65,72 65,72 65,72 64,75 64,75 64.75 64.75 64.75 64,75 72.90 72.90 72,90 72,90 75.84
65,84
40,45 40.4S 40:45 40.4~
72.72
72.77 72,77 72.77 72.77 72.77 72,77
78,84 ?8,84 84,84 78,84
Table 2. Impossible Problem solution pairs for different upper bounds up to 100. 32
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
Seating them at far separated tables, A addressed them from a central position. "Written on my piece of paper here are two distinct positive integers greater than two. Lying before you each is an envelope. Only the sum of these two integers is in your envelope, S, only their product is in yours, P. In a moment you may both~took at your numbers. The first of you who correctly names the two integers will receive a cash prize of $50,000, but I shall deduct $1,000 for every minute that elapses before you succeed. A guess will not be accepted because you must explain the logic used in reaching your answer. If your opponent or I can find fault with it, no prize will be awarded. All clear?" P and S nodded and after a moment A gave a sign and started his stopwatch. P and S then opened their envelopes, drew Out their numbers and began thinking. After about ten minutes S suddenly named the two integers. "Explain your reasoning," said A, stopping his watch. S then described her chain of inference, to which A and P raised no objection. A then held up the piece of paper to let P see that they were indeed the two numbers S had named. S then received about $40,000, as promised. What were the two numbers?
Solution to The Superimpossible Problem As previously, call the t w o u n k n o w n integers x a n d y, let p stand for the p r o d u c t a n d s for the sum; if there are only t w o distinct integers greater than t w o w h o s e p r o d u c t is p, then call p unique. We can a s s u m e P and S did their u t m o s t to a n s w e r the question as fast as possible because of the prize m o n e y diminishing rapidly with time. As m a t h e m a t i c i a n s their c o m p e t e n c e to reason a n d calculate fast is assured. N o t e that the timing in w h a t follows is less arbitrary than it m a y seem, a point to be g r a s p e d only t h r o u g h analysing the positions as seen t h r o u g h the eyes of the protagonists. Suppose p is unique b u t not large, say less t h a n a few hundred. P w o u l d then be able to factor his p r o d u c t with ease, and thus n a m e x a n d y virtually at once. Hence, a s s u m i n g a smallish s u m , S, w h o could then see that p is not large, will k n o w f r o m P ' s silence that p cannot be unique. Similarly, s u p p o s e s is one of its t w o lowest possible values, 7 a n d 8, which can be f o r m e d only b y 3 + 4 and 3 + 5, respectively. S w o u l d then be able to n a m e x a n d y instantly instead of taking 10 minutes. So s is greater than 8. Consider next the succeeding possibilities for s, in turn. The values involved are small, so that p w o u l d be small also. O b s e r v e that the time n e e d e d b y P or S to deduce a fact s h o u l d not be confused w i t h h o w long it m i g h t take us to infer the same. Suppose s = 9, w h i c h can be f o r m e d b y 3 + 6 or 4 + 5. T h e n p w o u l d be 18 or 20. But 18 a n d 20 are b o t h unique products. Therefore s cannot be 9. S u p p o s e s = 10, w h i c h is 3 + 7 or 4 § 6. Then p w o u l d be 21 or 24, the f o r m e r unique but the latter non-unique: 24 = 3 x 8 or 4 x 6. S k n o w s that if P h a d 21 he w o u l d be able to n a m e x , a n d y at once. Thus a silence of m o r e t h a n a m i n u t e w o u l d tell S he m u s t h a v e 24, a n d then she
w o u l d k n o w x a n d y at once. H o w e v e r , 10 w h o l e m i n u t e s elapse w i t h o u t S n a m i n g the t w o integers. Therefore s cannot be 10. S u p p o s e s = 11, which is 3 + 8 or 4 + 7 or 5 + 6. T h e n p w o u l d be 24 or 28 or 30. N o w 28 is unique, while 24 and 30 ( = 3 x 10 or 5 x 6) are not. P ' s silence will h a v e told S that p is not 28, but h o w could she decide b e t w e e n 24 a n d 30? S will consider P ' s situation. A s s u m e P has 24, w h i c h is 4 x 6 or 3 x 8. T h e n P w o u l d reason that s is 10 or 11. H o w e v e r , on considering 10, P w o u l d conclude as we h a v e done above, a n d so realise that S could t h e n n a m e x a n d y within a couple of m i n u t e s at most. So, as time ticks b y w i t h o u t S saying anything, after four or five m i n u t e s P could be certain that S m u s t h a v e 11 instead, and could then n a m e x and y. H o w e v e r , w e k n o w that P says nothing. Therefore p cannot be 24, a n d in the m e a n t i m e , S, w h o will h a v e been able to p u t herself in P ' s shoes a n d analyse the s ~ 10 case herself, can deduce this exactly as w e have. Thus, with S holding 11, and h a v i n g d e t e r m i n e d that p is not 24 or 28, she w m k n o w p can only be 30. So S n o w k n o w s that x a n d y are 5 and 6. The c o m p l e t e a r g u m e n t has been intricate, however, and caution will d e m a n d a careful re-check before speaking, in case of a n y mistake. This will take b u t a few m o r e minutes, bringing the total u p to a r o u n d 10, and t h e n S w o u l d be utterly confident. At this point she n a m e s the two n u m b e r s as 5 a n d 6. Can we be sure that this is the o n l y a n s w e r of its kind? I m a g i n e the competition had e n d e d differently w i t h P n a m i n g the t w o integers after a b o u t five minutes. This is a new, simplified puzzle. On retracing the a b o v e solution it can b e seen that P m u s t n o w h a v e 24, w h i l e S still has 11; the two u n k n o w n integers are then 3 a n d 8. The logic i n v o l v e d here o u g h t to ring a bell. The n e w p u z z l e is basically o u r old I m p o s s i b l e P r o b l e m w i t h a few details changed: the u p p e r b o u n d is gone, the lower b o u n d is n o w 3, not 2, a n d the t w o u n k n o w n n u m b e r s are defined to be distinct. Earlier w e noted that P w o u l d be able to tell that S cannot n a m e his p r o d u c t w e r e she to remain silent long ehough, rather than s p e a k i n g first at all. H e n c e S's "I see no w a y y o u can d e t e r m i n e m y s u m " can be excised, a n d still P w o u l d be able to say "I k n o w y o u r s u m , " after a n interval. The Superimpossible Problem arises from seeing w h a t S could deduce were P then not to say a n y t h i n g after all: in the a b o v e case, that P cannot h a v e 24, a n d thus m u s t h a v e 30, instead. The Superimpossible Problem is really a superstructure erected u p o n an u n d e r l y i n g Impossible Problem. H e n c e the question: Can we be sure that 5 a n d 6 is the only solution of its kind?, devolves to a similar question a b o u t the uniqueness of the solution to the u n d e r l y i n g Impossible Problem. The a n s w e r is yes, but I shall leave its proof w i t h the reader.
Buurmansweg 30 6525 RW Nijmegen The Netherlands THE MATHEMATICAL INTELL1GENCER VOL. 17, NO. 1,1995
33
Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions--not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the card where the famous conjecture was made, the desk where the famous ini-
rials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we inviteyou to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.
The Cornish Caveman Mathematician H.P. Williams The southwest edge of Bodmin Moor in Cornwall is wild and remote. It is Iess visited than other parts of the moor, as it is away from the major road through Cornwall. Nevertheless, it is an interesting and romantic area. There
Figure I *Column Editor's address: Mathematics Institute, University of Warwick, Coventry, CV4 7AL England.
are the Hurlers, two prehistoric stone circles. It was near here, on Twelve Mens Moor, thatJem Merlyn, the amiabIe rogue of Daphne du Maurier's Jamaica Inn, lived. There is also the Cheesewring, one of the most spectacular of the granite tots. Less known is that here, over 250 years ago, a recluse mathematician named Daniel Gumb lived in a cave (see Ref. 1). Daniel Gumb was a stonemason, born in 1703 in the parish of Linkinhorne about 4 miles away. He was introverted, and although both his father and grandfather were illiterate, Daniel was well read and acquired a considerable knowledge of mathematics and astronomy. He sometimes worked in the quarry, now long disused, beneath the Cheesewring. When he married the second of his three wives in 1735, he decided to make his home here. It is not known if this was to avoid the council tax or simply a reflection of his reclusive nature and an opportunity to observe the stars at night. He found a large slab of granite and dug out a 12-foot-deep cave beneath it. Here he lived with his wife, and eventually his children also, through the wind, rain, hail, and snow of Cornish winters on the exposed moor. Whether he continued to live here through his third marriage does not seem to be known. On the granite stone entrance to the cave he carved "D Gumb 1735." Mathematical diagrams were carved on other pieces of granite which made up the cave. In the middle of the last century the now-deserted cave was damaged by an extension of the quarry, but part of it was moved to a site nearby. The entrance, with its inscription, remains, together with the large granite roof. On the roof is a diagram reported as a "very ingenious method of proving the Theorem of Pythagoras (Euclid 147)"[2]. It is illustrated in Figure 1.
34 THEMATHEMATICALINTELLIGENCERVOL.17, NO. 1 ~ 1995Springer-VerlagNew York
Continued on p. 64
A Rejoinder to Hales's Article Wu-Yi Hsiang
In April 1994, I received a prepublication copy of the long article by Hales entitled "The status of the Kepler conjecture" [1] from Chandler Davis, Editor-in-Chief of the Mathematical Intelligencer, who informed me that it would appear in vol. 16, no. 3. This enabled me, for the first time, to inspect the mathematical content of certain objections, which have been circulated in the mathematical community in the form of rumours, to my article on the sphere packing problem and the proof of the Kepler conjecture [2]. In a letter signed by J.H. Conway, T.C. Hales, D.J. Muder, and N.J.A. Sloane, published in vol. 16, no. 2 of the Mathematical Intelligencer, the correspondents announced that their objections to my article would be articulated in the forthcoming article by T.C. Hales. Thus, the objections raised in Hales's article will henceforth be referred to as their objections in this rejoinder. 1 After a close inspection of the mathematical substance of their objections, and an item-by-item "reality check" between my article and their objections to it, I can now make the following reply. I will focus on the mathematical statements which they have made or attributed to me. These objections are concerned with the following two aspects of my work: (i) the geometry of local extensions and volume estimation (see [2], w11) and (ii) the geometry of core packings with 13 (or more) close neighbors (see [21, w Their objections can be briefly summarized as follows: 1. A fake "counterexample" to one of my volume estimates which grossly violates the explicitly stated condition for that estimate to be valid; cf. Objection 1 and Response, below. 2. They claim that a certain configuration ([1], Diagram 8) demonstrates a "methodological error" on
my part in carrying out my volume estimations. In reality, it only demonstrates that they have failed to comprehend the geometric setting entailed in the discussion of [2], Lemma 11; cf. Objeetion 3 and Response. . They coin the phrase "critical case analysis" to portray some of my proofs as mere "numerical checking" of a few experimental cases. Actually, the reductions of general estimates to critical cases are based on various simple geometric deformations; cf. Objection 7 and Response. . A few fallacious statements of their own are created as though I had relied on them in my proof, such as the one indicated by [1], Diagram 9 and the "Hole-Fitting Hypothesis." In fact, none of them has anything to do with the proofs found in [2]; cf. Objections 4 and 5 and Response.
1 They had not informed m e either about their letter or about Hales's article. I learned of the existence and scheduled publication of both after I wrote an inquiry to Prof. Davis on March 12, 1994. He finally sent me a copy of Hales's article which arrived on April 12. THE MATHEMATICALINTELLIGENCERVOL.17, NO. 1 (~1995 Springer-VerlagNew York 35
Response: The above-mentioned estimate is valid under a
a
m ~o
9
a clearly stated condition [2], p. 810), namely,
eQ.@Q
9
;
9
."
9
"
9 9
;
\I"
O))~D
O0
o
oo~
q) a ) Q ~ q ) e ~ i ) D
Figure 1. b = ~r/3 + 0.1133655 4~ chord length = 1.0965. O n t h e G e o m e t r y o f Local E x t e n s i o n s a n d Volume Estimation
Recall that the locally averaged density defined in [2], w is a specific weighted average of the local densities of a central sphere and its close neighbors. (Such a cluster is called a core packing.) Geometrically, such a local invariant depends on the structure of a double-layer local packing consisting of the central sphere, its neighbors, and the neighbors of its close neighbors. This doublelayer local packing is obtained from the core packing by adding a second layer consisting of their neighbors, thus making the local cell of every member of the core packing well defined. For each individual close neighbor, this extension of the core packing amounts to a local extension of a given partial surrounding to a full-fledged local packing. The proof of Kepler's conjecture can be reduced to proving that 7r/v/~ is the maximal value for the locally averaged density. This is Theorem 2 of [2]. The proof of this theorem is a kind of collective volume estimate of the local cells defined by such a double-layer local packing. Specifically, we show that the weighted averaged of their volumes is at least 4v~. Their objections on this aspect of my proof, and my response, are as follows. Objection 1: They claim to have a counterexample to my estimate in Example 1 [2], w p. 810. Their example is illustrated by [1], Diagram 7, or, by means of spherical polar coordinates of the touching points, by Figure 1 here. The radii of the two circles are 7r/3 and 2.04, the center and the five points on the circle of radius 7r/3 are the touching points (on SA) of its given partial surrounding, whereas the five points on the circle of radius 2.04 and the antipodal point of the center are the touching points of the add-on spheres in the local extension. This, they claim, is a counterexample to the estimate in Example 1 of [21, w 36
T H E M A T H E M A T I C A L I N T E L L I G E N C E R V O L . 17, N O . 1,1995
If the total amount of buckling effect (i.e. the buckling heights between the add-on spheres and those spheres of the given partial surrounding of SA created by the local extension of SA) is at most equal to 0.1, then... The first thing is, of course, to check whether their "counterexample" satisfies this explicitly stated condition, which requires the total buckling effect to be at most 0.1. In the example they give, there are 10 buckling effects between pairs of spheres corresponding to the pairs of points joined by the 10 intervals forming the star in Figure 1. The spherical distance b indicated in Figure 1 can easily be computed by means of the cosine law in spherical trigonometry, as follows: 71"
7I"
71"
cos b = sin ~ sin 2.04 cos ~ + cos ~ cos 2.04 = 0.398823237. The corresponding buckling height is, by definition, equal to one-half of the center-distance minus 1, which is exactly equal to the chord-distance minus 1, namely, b h = 2 sin ~ - 1 = 0.096518821, and, hence, the total buckling effect of their "counterexampie" is equal to 10h = 0.96518821. which is much larger than the maximum 0.1 required by the stated condition. Thus, this is no counterexample at all. Let me quote their comment following [1], Diagram 7: "Yet it is hardly surprising that counterexamples can be found . . . . " W h a t is not so surprising, perhaps, is that such a fake counterexample can only be manufactured by a wholesale disregard and gross violation of explicitly stated conditons. Second, let us look at how this fake counterexample has been presented as if it were an authentic one. In the paragraph immediately preceding "Here is the counterexample . . . . "the crucial condition requiring the total buckling effect to be at most 0.1 is conspicuously absent in their misrepresentation of that estimate. Fortunately, my article [2] is now published, so that truly concerned readers have a source for reality check. However, it is not unlikely that many readers will fall victim to such a misrepresentation and be misled into believing that this is a genuine counterexample; one takes for granted the faithfulness of the representation when such serious criticism is being raised against a mathematical proof. Third, the reason why it suffices to consider those local extensions with only a small amount of buckling effects is clearly discussed in [2], w For example, the existence of just one single local extension of the one described in
their "counterexample" will bring with it an additional volume of at least 0.9, which certainly will render the proof of Theorem 2 in such a situation trivial! Objection 2: In [1], note 11, they question the validity of the remark on L e m m a 5 [2], p. 757, namely that the volume of an hi-slab is at least equal to h i even for 0.052 < hj <_0.09. They conclude note 11 with
Perhaps the slabs do have volume at least hi, but this does not follow from Hsiang's stated method of estimation. Response: The method of estimation for hj > 0.052 is discussed in the last paragraph of the proof of Lemma 5 ([2], p. 757). The computation required in carrying out the estimation indicated there goes roughly as follows. Geometrically, there is no longer sufficient room to surround such a pair {So, Sj} by five neighbors touching both spheres if h; > 0.051462. Therefore, the tighter ones (i.e., with smaller volume for the hi-slab than otherwise) will have four neighbors touching both and a fifth one touching So and placed as close to Sj as possible. It is not difficult to show that the tightest one is the arrangement of five touching neighbors of So with their five touching points {Ai, 1 < i < 5} satisfying
A~Aj = arccos ~1 -+-h, i AIA2 = A2A3 . . . . .
1
A5A1 = - . 3 Set r = arccos[(1 + hj)/2], c~ = LA1AjA2, "/ = /A1AiAs, and b = AjA5. Then the base of the hi-slab
is the (Euclidean) pentagon indicated in Figure 2 with area given by r c~ r "y A(hj) = 3 tan 2 ~ tan ~ + 2 tan 2 ~ tan ~ + (ll + 12)~, where b
=tan ~-tan
r "~ 11 = tan ~ tan ~,
r
~,
12 = 11 - 6 cot %
Straightforward computation will show that A(h d) is a decreasing function of hj and A(0.09) = 1.22896835. Thus, it is easy to see that the most critical case for the verification of the remark is w h e n h i = 0.09, and the lower b o u n d for the volume of the hi-slab is 0.0905576 ... for hj = 0.09. Contrary to their assertion, the lower bound of h d follows readily from the method of estimation indicated in that remark [2], p. 757; moreover, the computation above is almost identical to that in the proof of [2], Lemma 5. Objection 3: Under the heading "Diagram 8, A methodological error" [1], they exhibit a configuration with 11 close (in fact, touching) neighbors and another non-close neighbor with center-distance barely exceeding 2.18, call it 2.18 + e. They claim that although the volume of the local cell is somewhat larger than 4v~, it cannot be accurately estimated by the techniques developed by Hsiang. They proceed to interpret w h a t should be Hsiang's method of volume estimation, beginning with "We are instructed to . . . . " and end up with the estimate 5.6131. Then they solemnly declare,
The importance of this example is that it escapes the methods and techniques that Hsiang has introduced to solve the problem 01). His claim to have used these method to establish a bound is incorrect. It means that he really offers no proof of the lemma, and that his "set of convenient tools for the proof of Theorem 2 [the Kepler conjecture]" remains unsubstantiated(12). (Their note 11 is exactly Objection 2.)
Figure 2. Minimal base of the h-slab; radius = tan ( r / 2 ) , arccos [(1 + h)/2].
r =
Response: A great deal of effort must be expended to distort m y methods and techniques for volume estimation to produce such a crude lower b o u n d as 5.6131, thus enabling them to proclaim my so-called "methodological error" and to say that "he really offers no proof of the lemma . . . . " In fact, one m a y simply estimate the (0.09 +D-slab by 0.09 and the volume of the tangent polyhedron by the lower b o u n d 2011 - 2 cos (2~r/5)] tan (7r/5) = 5.550291. This would already provide a lower bound of 5.640291 for any core packing of the type proposed by them. If one applies the volume estimation lemmas of [2], w in a slightly more refined way, the above lower b o u n d of 5.640291 can be improved to 5.65. Moreover, the geometric setting of Lemma 11 [2] is m u c h more restrictive, and, THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
37
hence, it is easy to obtain a lower bound exceeding 4v~ (by a considerable amount), the restriction being that a
very specific type of partial surrounding is already given. Let s be a local extension of the SA considered in Lemma 11 which consists of 11 close neighbors and another non-close neighbor with center-distance (2.18 +c). Let us consider the case where the six add-on close neighbors are all touching ones. (This case is clearly more critical than others.) Then the direction vectors of the above seven add-on spheres are within the close vicinity of the complementary region indicated in [2], Figure 19; moreover, the subconfiguration spanned by them is a small deformation of the one indicated in Figure 20 of [2]. Therefore, the associated configuration of such a local extension of 12 neighbors is a close deformation of the 5D-type, and, hence, the volume of the associated tangent polyhedron has an easy lower bound of 4 v ~ - 0.03. Thus, the volume of the resulting local cell of such a local extension is at least 4 v ~ - 0.03 + 0.09 = 4 v ~ + 0.06. What produced this "nonapplicability" of the techniques and methods developed in [2] to the proof of Lemma 11 was the rigid preconceptions which had led them to ignore the geometric setting of that lemma, and my "methodological error" is a result of their failure to comprehend the geometry entailed in the discussion. O b j e c t i o n 4: They created the terminology "The Hole-
Fitting Hypothesis" and then proceeded to construct a counterexample to that hypothesis in the case of circle packings. They claim that I invoke that hypothesis in the proof of Case I3 of Theorem 2 ([2], pp. 818-819), then assert, "But if the principle is not sound, he is not justified in invoking it at all." Response: This so-called "Hole-Fitting Hypothesis" is purely of their creation and has nothing at all to do with my proof. I told Mr. Hales in 1992 at the University of Chicago that I did not use such a hypothesis and even showed him that one could not fill all the small holes in the extension of the fcc core packing which achieves the optimal locally averaged density ~r/v/~. The false accusation that I somehow rely on such a hypothesis in m y proof of Theorem 2 must be the result of a gross misunderstanding of my article [2]. Let us consider the specific case of a 5D-type core packing (Case I3, pp. 818-819). The star configurations of the touching points of the 10 touching neighbors are of the type (G, G, •, •). Let the sphere SA be I of the 10 touching neighbors, with A as the touching point. Thus, the core packing defines a partial surrounding of SA, and the complementary region of such a partial surrounding on the surface of SA is indicated in Figure 3. The vertices B1 and B2 (resp. C1 and C2) correspond to the touching points of the spheres fitting into its adjacent big (resp. small) holes. Note that on SA the point A (which is the antipodal point of the center of the figure) is a 4-fork vertex with angular distribution (~, c~, fl, fl), and the as-
38
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
sociated configuration of a local extension with 12 close neighbors always contains it least two 6-fork vertices. It is not difficult to see that such a local extension with at most 0.05 additional buckling heights must be a small deformation of either the 5[]-type or the hcp-type configuration, and that in either case the configuration must contain points in the vicinity of B1 and B2 (indicated by the shaded regions). In other words, such a local extension always includes spheres on top of both big holes. If the spheres on top of the large holes are non-fitting, the buckling effect created by such a local extension will, of course, contribute a much larger volume-increment than any possible amount of volume-decrement toward the collective estimate. Therefore, it is indeed the case that fitting five spheres to the five big holes of a 5D-type core packing will produce smaller total volume than otherwise. Moreover, such a local extension, whose configuration is a small deformation of the 5 []-type (resp. the hcp-type), requires the touching points to be placed in the vicinity of both C1 and C2 (resp. M). The 5D-type will, thus, produce a considerably smaller local cell for the remaining two close neighbors than the hcp-type. Therefore, to fit 10 spheres to its 10 small holes will again produce a smaller total volume than otherwise. Clearly then, this argument is based on the special geometric structure of 5[]-type core packing and has nothing whatever to do with their "hole-fitting hypothesis"!
Remark: Although the proof of Theorem 2 involves the collective volume estimate of a cluster of local cells, the individual estimate of the volume of the resulting local cell of a local extension of a given type is always much easier than that of the nonrestrictive case encountered in the proof of Theorem 1. Note that the given partial surrounding already imposes severe restrictions on the geometric setting, with much smaller freedom than the nonrestrictive setting.
t~
C~ ,.".,.:-,...-,',~
,;,;~')~.C2
Figure 3. Complementary region on SA seen under stereographic projection: Members of the local extension can only touch SA within this region.
On the Geometry of Core Packings with 13 Close Neighbors The classical problem of 13 spheres, originated in a discussion in 1694 between Gregory and Newton, is to determine the maximal number of spheres that can touch a central sphere. Newton believed that the number should be 12; Gregory thought that 13 might also be possible. The maximal number of touching neighbors was finally proved to be 12 in 1953 by Schfitte and van der Waerden [3]. However, the mere statement of the impossibility of 13 touching neighbors is of little use for the purpose of density estimation, because there is clearly very little difference between touching and almost-touching neighbors in the quantitative problem of density. Recall that the buckling height between two neighbors, So and Sj, is defined to be hj = 1OOj - 1, that is, half of the gap between them; this is an exact quantitative measure of the "nontouchingness" between the two spheres. Thus, the sum total of the buckling heights contained in a core packing is a quantitive measure of its total nontouchingness. What is needed for the proofs of both Theorems 1 and 2 of [2] is a sufficiently good lower bound for the total buckling height of a core packing with 13 (resp. 14 and 15) close neighbors, namely, 0.325 (resp. 0.56 and 1) [although a smaller lower bound of 0.13 (resp. 0.25 and 0.34) already suffices for the proof of Theorem 1]. There are three closely related geometric problems: (i) The problem of whether it is possible to have 13 touching spheres. (ii) What is the smallest spherical surface on which it is possible to arrange 13 touching unit spheres? (iii) Lower bound for the total buckling height of core packings with 13 close neighbors. It is natural to reduce these problems to problems in spherical geometry by looking at the spherical shadows cast (i.e., the solid angles of the cones subtended) by the neighboring spheres. Then, the first two problems are reduced to problems concerning circle packings on the unit sphere, namely:
A~
m 0
A6
A~
A,
Figure 4. Area-decreasing sheafing deformation of a 6-fork star. (i) the area estimate for individual triangles and for the total area of a 6-fork star, that is, Sublemmas 3 and 5; (ii) the control of combinatorial types of ~pherical configurations with 13 vertices, that is, Lemma 6. The advantage of such a proof by area estimates and combinatorial analysis is its flexibility and direct adaptability to the general case of nonuniform buckling heights. Let us see how it works. Set a0 = 2 arcsin(1/2.18) = 0.9531 and
l(hi, hi) = arccos 7r
3
1 + 2(hi + hi) + (h~ + h2) 2(1 + hi)(1 + hi)
hi + hj
v~
mod terms in (h 2 + h2),
A(hl, h2, h3) = area of the triangle with side-lengths/(hi, h2), etc. = 2 arctan v5~
~ ~ 2 ( h 1 q'-
h 2 -]- h 3 )
mod terms in (h 2 + h22+ h~).
(i) Is it possible to pack 13 circles with radii 7r/6 on (the surface of) the unit sphere? (ii) What is the maximal value of r for which 13 circles of radii r can be packed onto the unit sphere?
In applying area estimation to check the validity of Lemma 8 for the case of nonuniform buckling heights, one has the following modifications, which are clear consequences of the proofs of Sublemmas 3 and 5:
On the other hand, the problem in spherical geometry corresponding to the third problem is much more complex and irregular. It involves 13 circles of radii varying in the range between arcsin (1/2.18) and 7r/6, to be placed on the unit sphere so that a- limited amount of overlapping is allowed. In order to handle this problem effectively, a fundamental improvement in methodology is required. The following method of area estimates and combinatorial analysis was, developed in w of [2] mainly for the purpose of proving Lemma 8; it consists of
(i) The area of /XAIA2A3 is at least equal to /X(hl~ h2~ h3), where hi is the buckling height of Si with OAi as the direction of OO~. [Note that S(E) is a0-saturated and, hence, the circumradius of every polygon in S(E) is less than a0.] (ii) The proof of Sublemma 5 uses the simple type of area-decreasing deformation indicated in Figure 4, which localizes the deformation to a quadrilateral formed by consecutive boundary vertices, such as hA1A2A3A4, by shearing it further away from the cocircular shape. THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1, 1995
39
,a2
A,
Cs
,4
/9,
Their main objections to the proof of Lemma 8 of [2] can be summarized as follows.
C=
O~
, "~,
A=
C3
;2
Figure 5. A 7-fork star with exactly four long forks and its complementary region; the external angles at B1, B2, and B3 are considerably smaller than 7r.
Therefore, the same proof shows that those 6-fork stars of minimal areas with respect to a given set of constraints on edge-lengths such as AiAj >_l(hi, hi) always occur at the most lopsided ones which consist of two triangles, one buckled quadrilateral and one cocircular quadrilateral, all with minimal side-lengths determined by the distribution of buckling heights, namely, AiAj = l(hi, hi) except for the diagonals of the two quadrilaterals. In order to check the validity of Lemma 8 for the case of nonuniform distribution of buckling heights by the same kind of area estimates, the cases which actually require some computation are those where the total buckling heights of 0.325 available are highly concentrated at the vertices shared by the two 6-fork stars. This situation is more critical than others because such concentration will cause the maximal amount of area-decrement for the given amount of buckling heights. Referring to Figure 7 in [2], this situation will occur, for example, when the distribution of buckling heights is 0.09 at A0, A3, and A4 and 0.055 at A1 in the case of Figure 7(i), or when the total buckling height (0.325) is distributed solely among the four vertices []A1C1A2C2 in the case of Figure 7(ii), (iii), and (iv) and cM1C1A2C2 is cocircular. These are among the most lopsided distributions, and they will make the area of the double-star considerably smaller than the corresponding one with uniform buckling heights h = 0.0316. However, such area-decrement is more than adequately compensated by
the increment of the area estimates for its complementary subconfiguration, because of the absence of buckling in the remaining vertices. Straightforward computations will show that the total area of S(G) always exceeds 4~r by a larger margin than that of the corresponding case of uniform buckling height h = 0.0316 which is exhibited in [2]. This is, of course, to be expected, as 0.325 is 25% smaller than 13h. The computations involved in the verification will consist of rather simple spherical trigonometry, together with straightforward accountings of area estimates of the same kind as those exhibited in [2] for the uniform case, where the averaged buckling height takes the higher value 0.0316 instead of 0.025 in the nonuniform case. We refer to [41 for the detailed computations in the case of nonuniform buckling heights. 40
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
Objection 5: Under the headings "Reduction to critical cases" [1], p. 53) and "Nonexistence of contractive deformations" [1], p. 54), their objections to my proof of Sublemma 7 [2], w are that:
(i) I made a fallacious claim as indicated in Diagram 9 of [1]. (ii) I must be relying on the existence of "sizedecreasing" (or contractive) deformations. Several pages of [1] (pp. 53-56) are then devoted to proving the nonexistence of such deformations by an example. Response: First of all, nowhere in [2] did I make the
fallacious claim which they have attributed to me. It is strictly their creation (which was somehow also adopted by I. Stewart in [5]), having nothing to do with my proof of Sublemma 7. Second, the proof of Sublemma 7 made no use of contractive deformation of the 7-fork star. In fact, it is not the area of the 7-fork star which is crucial for the proof of Sublemma 7. What is crucial is the size and shape of the complementary region to the union of the eight discs of radii 0.98 centered at the vertices of the 7-fork star, as indicated in Figure 5. In that figure, C 1C2, C2C3, C3C4, C4C5, and C5C1 are circular arcs of radius 0.98 centered at/?3, A4, A1, A2, and A3, respectively. It is not difficult to see that a 7-fork star (of edge lengths at least 0.98) contains at least four long forks, and the only case for which the proof is not altogether obvious is when the four long forks are separated in a (1, 1,2) manner by the three short edges of the 7-fork. Indeed, it is quite easy to reduce the proof further: to consideration only of those 7-forks with exactly four long forks with lengths >0.98. Such a reduction to the critical cases is very easy to justify, as the elongation of any short edge is necessarily accompanied by an elongation of the long forks. Here I would like to remark that the key points in the proof of Sublemma 7 are as follows: (i) The lengths and directions of the four long forks which are determined by the distribution of the central angles. (ii) The shape and size of the resulting complementary region. (iii) The distance between P and C3 is always much smaller than 0.98. Objection 6: In note 8 [1], they complain that Sublemma
7 is proved with the separation of 0.98, and, therefore, it cannot be applied to the proof of Lemma 8, in which cases might occur with a few edges as short as 2 arcsin (1/2.18) = 0.9531.
Response: Apparently, they have totally disregarded the remark immediately following Sublemma 7 and the simple fact that the presence of just one single edge of length close to 0.9531 will use up a buckling height of 0.18 or more for the pair of close neighbors corresponding to the vertices linked by such a short edge, whereas two such edges will use up a buckling height of at least 0.27 for the corresponding triple. Consequently, only 0.145 (for a pair, at most 0.055 for a triple) total buckling height is left to be distributed among the other 11 (resp. 10) close neighbors, thus forcing a large number of short edges very close to ~r/3, which are considerably longer than 0.98. The nonexistence of 7-forks in the configuration of a core packing of 13 close neighbors with total buckling height 0.325 is clearly implied by the proof of Sublemma 7. Objection 7: Under the heading "Critical case analysis," they claim that many essential inequalities found in my article [2] are based on "critical case analysis" rather than rigorous and generally valid argument. Response: In the lower bound estimate of the total buckling height of core packings with 13,14, or 15 close neighbors, the method of area estimation and combinatorial analysis in [2] is, of course, based on the following two essential inequalities: (i) The area of every triangle (resp. quadrilateral) in the spherical configuration of a saturated core packing is at least G(hl~h2~h3) [resp. the area of the cocircular quadrilateral with side-lengths l(hl, h2)~ l(h2~ h3), and so on ([2], Sublemma 3). (ii) The area of a 6-fork star configuration is minimal only when it contains exactly two edges longer than the constraints l(h~, hd) and one of these two edges is the diagonal of the cocircular quadrilateral described in (i) ([2], Sublemma 5).
,"
\
/
Figure 6. Area-decreasing deformation of triangles with the same circumradius:' b' < b < c, circumradius < a0 = 2 arcsin (1/2.18).
A natural way to provide clean-cut arguments for these useful area estimates is to reduce the proofs to more critical cases by means of simple area-decreasing deformations. For (ii), this can be done with the shearing deformation indicated in Figure 4. For (i), the deformation used is indicated in Figure 6. Because the area is always decreased under these deformations, it is easy to see that the general inequalities involved in (i) and (ii) follow from the computation given in [2] for the critical cases. Their failure to see the existence of such simple geometric reductions has led them to label such proofs as "critical case analysis," meaning that they are merely "empirical verification based on limited experimentation" [1, p. 48]. In fact, the geometric deformations I used for such reductions in [2] are all intuitively quite obvious and technically easy to verify; they are well-known facts in spherical geometry. Obj ection 8: Their major objection to the proof of Lemma 8 was summarized imthe last sentence i~ "Historical grounds for skepticism": On a problem of considerable historical significance it is not sufficient for Hsiang to argue that because the case he tried works by a comfortable 25% margin, things must also work in general. Response: Indeed, this is a problem of considerable historical significance and the quantitative lower bound of 0.325 for the total buckling height of 13 close neighbors goes considerably further than the purely qualitative result on the impossibility of 13 touching neighbors. The first correct proof of the impossibility of 13 touching neighbors, given in 1953 by Sch/.itte and van der Waerden [3], was a remarkable achievement. However, a new approach and more powerful techniques are needed to obtain such a lower bound on the total nontouchinghess. The method of area estimation and combinatorial analysis was developed in [2], w mainly for this purpose. Observe that the area estimates for individual (spherical) triangles and for the total area of a 6-fork star, as well as the control on the combinatorial possibilities of spherical configurations with 13 vertices [2], Lemma 6, are valid for the case of nonuniform buckling heights in just the same way as the uniform case. In the case of a lopsided distribution of buckling heights, the area estimates for those triangles (and 6-fork stars) with averaged buckling height exceeding 0.0316 will, of course, be smaller than the corresponding estimates in the case of uniform 0.0316-buckLing. But for such a lopsided distribution, many other triangles or 6-fork stars must have averaged buckling heights considerably less than 0.025. For our purposes, 0.025 is significantly smaller than 0.0316, and the excess they contribute toward the total area estimate is more than sufficient to offset the deficit. Hence, the area estimate in the nonuniform case is, in fact, considerably larger than in the uniTHE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
41
form case. To check the details, it is only necessary to carry out some straightforward computations in spherical trigonometry, together with some simple accounting (cf. [4], Lecture 10).
Some More Reality Checks Over the past 2 years, there have been quite a few fallacious statements that were falsely attributed to me as my "elementary error" For example, some of them even got published in an article by Stewart [5] and then quoted in Hales's article as if they were established facts. The following is quoted from [5]: Hsiang's earliest preprint definitely contained errors. For instance, he claims that if the sides of a given triangle are larger than the corresponding sides of a second triangle, the first must have the larger area. However, a moment's thought shows that this is untrue. An equally fallacious claim made by Hsiang was that if several objects do not fit into a particular region, then they cannot fit into a region of smaller area. I believe the originator of the former statement is J.H. Conway and that of the latter is T.C. Hales. In any case, they are not my statements but are created by them perhaps to explain away their own misunderstanding of my earlier preprints. In my first short preprint on sphere packings, I stated the area estimate of Sublemma 3 [2], p. 766 without proof. It is, after all, just a simple geometric consequence of the fact that the circumradii of triangles in S(N) are bounded above by the given separation. Here again, their own misunderstandings were circulated as my elementary errors, and easily provable statements are tortured into fallacious statements by quoting them out of context or, in some cases, by outright misrepresentation. As for the second fallacious claim attributed to me in Stewart's article, nowhere did I ever use anything like it in my proofs. As it has already been pointed out in the Response to Objection 5, it is perhaps the result of misreading of the proof of Sublemma 7 of [2].
Conclusion This then is an item-by-item reality check on the mathematical content of their objections to my article [2], on which they have based their assessment on the status of the Kepler conjecture. It should now be clear from the responses that their objections are without any mathematical foundations. Let me conclude this article with a few remarks on the proof of Kepler's conjecture. (i) Although the definition of close neighbors to be spheres with center-distance at most 2.18 is not canonical, it plays a useful role in simplifying proofs in many places. (ii) The locally averaged density introduced in [2] w is a specific weighted average of the local densities of a central sphere and its close neighbors. It enables us to reduce the proof of Kepler's conjecture to the optimal estimation of such a local invariant (see [2], Lemma I and Theorem 2). 42
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
(iii) Lower bound volume estimation of the local cells in various geometric settings is, of course, the basic technique involved in the optimal estimations of both the local density and the locally averaged density (see [2], Theorems I and 2). A system of volume estimation techniques tailored for this purpose are developed in w1675, and 6 of [2]. (iv) The cluster of spheres consisting of a central sphere and its close neighbors is called a core packing. The geometry of core packings with 12 or 13 close neighbours are essentially the only ones whose local geometry plays a critical role in the proof of Kepler's conjecture. Techniques in spherical geometry are developed in order to analyze the relevant aspects of the geometry of these types of core packings (see [2], w167and 8). (v) Technically, the volume estimation of the resulting local cell of a given type of local extension is much simpler than that involved in the optimal estimation of the local density when there is no restriction on the local extension at all. Here, the control on the buckling effect and the understanding of the geometry of core packings with 12 close neighbors (see [2], w plays a major role. (vi) Kepler's conjecture is a natural and fundamental problem about three-dimensional Euclidean space (the space in which we live) which involves the arrangement of infinitely many identical spheres. It is natural that spherical geometry should play the major role throughout the proof of Kepler's conjecture. In retrospect, I find my understanding of spherical geometry considerably enriched through working on this problem. I think a truly interested and sufficiently patient reader of [2] will not only understand the proof of Kepler's conjecture but also share this valuable experience. Spherical geometry is the "orthogonal part" of Euclidean geometr~ where many of the deeper properties of Euclidean space reside. One may say that the study of the sphere packing problem enables us to understand E 3 better The techniques in spherical geometry developed for the purpose of proving Kepler's conjecture may well find further use in the geometry and physics of solids.
References 1. T.C. Hales, The status of the Kepler conjecture, Mathematical Intelligencer 16 (3) (1994), 47-58. 2. W.-Y.Hsiang, On the sphere packing problem and the proof of Kepler's conjecture, Int. J. Math 4 (5) (1993), 739-831. 3. Sch/itte K. and B. L. van der Waerden, Das Problem der dreizehn Kugeln, Math. Annalen 125 (1953), 325-334. 4. W.-Y.Hsiang, Lectures on the geometry of spheres, Department of Mathematics, University of California, Berkeley (1994) (mimeographed). 5. I. Stewart, Has the sphere packing problem been solved?, New Scientist (2 May 1992), 16.
Department of Mathematics University of California Berkeley, CA 94720 USA
Number Mysticism in Scientific Thinking Irving M. Klotz
For millennia, human beings have been tantalized by the idea that integers and patterns of numbers carry deep hidden meanings. They have searched assiduously for these cryptic insights. Numerological analysis has been embraced ardently in mythology, religion, and the sciences. I want to focus particularly on the fascination with integers during the evolution of concepts of molecular structure and behavior of protein macromolecules, an area in which I have lived for half a century. To place these in a broader perspective, however, I shall start with a sampling of number mysticism in other fields of knowledge.
the number corresponding to the name of the former is 1276, and to that of the latter 1225; "obviously," the larger number must be the dominating one. In the middle ages, Jewish Kabalists and Christian theologians developed
Number Lore in Mythology and Religion Let us recall first some integers of religious import [1]: 3 is a prime number in theology; 7 is the number of verses in the Fatiha in the Koran; 40 is the number of days of Moses's sojourn on Mount Sinai. Numbers up to 60 were associated with each of the Babylonian gods. An especially highly developed form of number mysticism is gematria [1, 2]. As every letter of the Greek, Latin, or Hebrew alphabet represents a number as well as a sound, the letters of a word can be summed to give the "number of the word" and, hence, to provide a new insight into hidden meanings of the word. The ferreting out of the import of such numbers became a highly developed art. For example, the victory of Achilles over Hector in Greek mythology was ascribed to the fact that THEMATHEMATICAL INTELLIGENCER VOL.17,NO.1(~)1995Springer-VerlagNewYork 43
the art of interpretation of gematria to increasing degrees of sophistication. In Christian theology, the n u m b e r 666, that of the Beast in the Book of Revelation, had a special significance (the Antichrist) and was used repeatedly to assail individuals or groups, Christian as well as nonChristian. For example, Pope Innocent IV, whose Latin name corresponds to 666, was branded the Antichrist by his opponents, and Luther's enemies pointed to the numerical equivalence of his name to 666. Such beliefs are still pervasive. Within the past year, the "secrets of 666" were revealed to me in a pamphlet provided by the Full Gospel Illinois Church in Chicago. "And he causes all, the small and the great, and the rich and the poor, and the free man and the slaves, to be given a mark on their right hand, or on their forehead, and he provides that no one would be able to buy or to sell, except the one who has the mark, either the name of the beast or the number of his name. Here is wisdom. Let him who has understanding calculate the number of the beast, for the number is that of a man; and his number is six hundred and sixty-six" (Rev. 13:17-18)
Soon money will be useless and in place of it, the mark will be put on the forehead or on the right hand, and this will be the only way to buy and sell. Bar codes of U.P.C. is the 666 mark and is already all over supermarkets, hospitals, and libraries. We beg you dear friends. Do not ever receive the 666 mark! The New Age religions and mythologies also welcome numerological analyses. I have been especially charmed by one such effort that draws on structural information about protein molecules. These very large molecules (molecular weight of ~-,105 if that for a hydrogen atom is set at 1) are constituted of long chains of linked smaller units, the amino acids (molecular weights ,,~102). There are 20 different natural amino acids that can appear in each position of a protein chain. For very m a n y proteins the precise linear sequence of amino acid residues in the chain(s) has been determined, and this voluminous information has been assembled in a data bank of protein sequences. Thus, one can raise mathematical questions such as the following: What is the probability of finding in the encyclopedic data bank the sequence of linked amino acids (Glutamic)(Leucine)(Valine)(Isoleucine)(Serine)?
(1)
The following a r g u m e n t seems eminently appropriate. As in each position there are 20 choices, and each position is independent, the probability must be
observed frequency of 1.5 x 10 -4 [3]. Thus this pentapeptide appears about 1000 times more frequently than expected by chance [Eq. (2)]. Obviously, there must be something significant encrypted in this result.
Some
eccentric
ideas in science seem immortal.
They do not die, they do not even fade away.
The h i d d e n message was uncovered when the standard one-letter notation was inserted [3] in place of the full name of each of the amino acids in the pentapeptide. (The one-letter code is: Glutamic = E: Isoleucine = I; Leucine = L; Serine ~ S; Valine = V.) As the reader can readily verify, in place of Eq. (1) we obtain ELVIS;
observed frequency = 1.5 x 10 -4.
(3)
This remarkable discovery prompted the observers [3] to formulate the following conclusion: Since that fateful day of 16 August 1977 when Elvis Presley, considered by fans the world over as "the King," passed on, there have been many attempts to uncover evidence that this rock and roll legend is still among us. For the most part, these efforts have been conducted in a haphazard manner and quite frankly have lacked credibility. Elvis sightings in shopping malls, doughnut shops, and aboard alien space craft have yet to be properly documented. We believe this report is the first credible evidence that "The King" is still among us. Numerology
in the Physical Sciences
Let us turn n o w to mainline science. In the spring of 1899, Max Planck published his revolutionary paper that introduced the constant h, n o w called the q u a n t u m of action. At that time, he was not aware of the astonishing implications of this Planck constant [4]. What actually excited him was that h, together with the velocity of light c and the gravitational constant G, offers the possibility of establishing units for length, mass and time which are independent of specific bodies and which maintain their meaning for all time and all civilizations, even those which are extraterrestrial and non-human; constants which therefore can be called "fundamental units of measurement." Their values are
(1)
(1/
(1/
(11(1)=3x10-7.
(2)
Planck length = (Gh/c3) I/2 Planck time = (G/r
Fast algorithms exist to scan the encyclopedic data bank in search of the pentapeptide sequence shown in Eq. (1). It occurred 4 times in the total population of about 26,000 pentapeptides known in 1991, i.e., with an 44
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
=
10 -33
cm,
(4)
1t2 = 10 -43 s,
(5)
Planck mass = ( h c / G ) 1/2 = 10 -5 g,
(6)
where G is the gravitational constant, h the Planck constant, and c the velocity of light.
The enormous disparity in scales of value of these constants has intrigued physicists for a century, and repeated attempts have been m a d e to explain the "hierarchy" [4]. For almost a century, theoreticians have also raised the question whether the fundamental constants of nature (e.g., velocity of light, mass of the electron, charge of the electron, Planck's constant, gravitational constant) are ind e p e n d e n t entities or are interconnected in some deep, subtle w a y still h i d d e n from us. This is still a central concern of particle physicists. There have also been some great minds, however, w h o have w a n d e r e d off into m o r e esoteric notions about the n u m b e r s associated with the fundamental constants. A prime example is A. S. Eddington, probably the most distinguished astrophysicist of this century [5]. A man of daring mind, he was very receptive to imaginative novel ideas. It was Eddington w h o in a dramatic fashion in 1919 brought Einstein and relativity to the attention of the public. Eddington found that a m o n g various combinations of the fundamental constants, one,
Table 1. Functions a Related to Velocity of Light Function
Numerical Value
Reciprocals
c
2.9976 x 10 l~
0.33356 x 10 -l~
2.9995 x 10
0.33339 x 10 -1
3.0009 • 109
0.33323 x 10 -9
2.9990 X 10 -14
0.33344 X 1014
2.9971 • 10 - 9
0.33366 x 109
2.9967 x 103
0.33370 • 10 -3
(8"-~1/2
-kin0/ rn0 2~re 27r~
3Gc 2 a Symbols
for and values of fundamental constants:
c = velocity of light = 2.99796 x 101~ cm s -1, h = Planck's constant = 6.547 x 1 0 - 2 7 erg s, m0 = mass of electron = 8.994 x 10 -2s g, = electronic charge = 4.770 x 10 -l~ absolute es units, G = gravitation consta4~t --" 6.664 x 10 -8 cl-n3 g-1 s-2
C2
c( h /2~r )
(7)
(where c is the charge of the electron), was a p u r e dimensionless n u m b e r with the value of 137. This exact integer he felt carried the secret of the universe, the link between physics and cosmology. Earlier in this century, the theoretical physicist Sommerfeld had discovered the fine-structure constant (c~)of atomic physics and its value (really that of 1/c~) was essentially 137. Eddington took this to be strong s u p p o r t for his ideas. Had he known, Eddington might have been intrigued by the fact that the four H e b r e w letters that spell Kabalah, the corpus of medieval Jewish mysticism, sum to the integer 137. With the enormous i m p r o v e m e n t in precision of modern determinations of the fundamental constants (uncertainties n o w reaching parts per billion), the accurate value of 1/c~, 137.0359895(61), is unquestionably not an integer. Even in Eddington's time, however, m a n y of his contemporaries were v e r y skeptical of his numerological approach. One of the famous hoaxes in the history of science derives from this skepticism. In 1931, the y o u n g Hans Bethe, today one of the eminent elder statesmen of physics, with two of his colleagues published a note in Naturwissenschaflen [6]. Using accepted terminology about degrees of freedom of electrons and protons, they created a succession of seemingly logical sentences leading to the following conclusion: Thus to reach absolute zero, we must remove (2/c~ - 1) degrees of freedom, associated with orbital motion. Hence for absolute zero we obtain To = -(2/c~ - 1) degrees Setting To = -273, we obtain a value of 137 for 1/c~.
Despite the correct g r a m m a r and syntax of the paragraph, there is actually no logical reason for the equal sign in the equation cited. Furthermore, To actually is -273.15~ which would give a noninteger value of 137.08 for 1/c~. A variant of Eddington's idea, in an outrageously foolish form, was published 50 years ago in a v e r y reputable chemical journal by a scientist n a m e d J. E. Mills [7]. He was fascinated by the n u m b e r 3 a n d set out to show that a large n u m b e r of products and quotients of fundamental constants have values v e r y close to this integer. A representative few are assembled in Table 1. For example, the velocity of light, c, is listed as 2.99796 (•176 the mass of the electron, m0, divided b y 27r times e, its electronic charge, is 3.0009 (x 109), and so on. M a n y tables of such concordances were assembled. These led Mills to the following conclusions. Ignoring the decimal point it will be at once evident that the agreement shown [better than one part per thousand [at that time]] between the figures given in each table could only occasionally be accidental . . . . All of the different series of figures are related one to another... [and] to the velocity of light . . . . This result is amazing. Mills n e v e r seemed to appreciate that these coincidances w o u l d not have existed if not for the French Revolution. A fundamental flaw in such concordances is that they are completely d e p e n d e n t on the units in which the constants are expressed. For example, the velocity of light expressed in cubits, li, kos, or miles is very different from the 3 x 10 l~ in centimeters per second. Curiously enough, after being m a d e aware of this, Mills took the following position: THEMATHEMATICALINTELLIGENCERVOL.17,NO. 1,1995 45
This may indeed be a grievous error. But if it is, the author is not to blame. The mistake was made by nature. All the author has done is to show that a numerical relation ~ictually exists between quantities of different dimensions. Nature, not the author, must be called on to explain why. Some eccentric ideas in science seem immortal. They do not die; they do not even fade away. T h e y merely lie dormant, or s u b m e r g e d in the collective scientific subconscious, and are revived often in a slightly different guise. Recently, half a century after Mills, a note appeared in Nature [8] relating the fundamental constants of physics to the n u m b e r 7 r - - o r rather, as that in itself doesn't serve, to various functions of ~r, such as 7r6 or 1 + 7r-2 + Ir -4. The a u t h o r was so elated b y his findings that he says I would state categorically that coincidence is ruled out. The n u m b e r ~r has bewitched amateurs for centuries, vexing m a n y of them b y its nonintegral value. There is a legend that an American state legislature once considered establishing b y law a value for ~r of exactly 3. This story has never been confirmed, but there is no doubt that the H o u s e of Representatives of the Legislature of the State of Indiana once unanimously passed a bill that in essence established a numerical value of ~r [9]. The value decreed is not explicitly stated; it has to be deduced from the following statement in Section 1 of H o u s e Bill No. 246 of the 1897 session, "A bill introducing a new Mathematical truth": Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side. Let us parse this statement into its parts. The first instructs us to take the ratio of the area of a circle to the square of a quarter of the circumference. Since Euclidean plane geometry is not declared invalid in the bill, it is presumably still accepted. So we m a y write Area of circle
7r 2
(Length of 1 / 4 circumference) 2
[1/4(27rr)] 2'
(8)
where r is the radius of the circle. The second part of the statement instructs us to take the ratio of the area of an equilateral rectangle, that is, a square, to the square of one side, a ratio which is unity, so we m a y write Area of square = 1. (Length of one side) 2
(9)
H o u s e Bill No. 246, which passed the H o u s e of Representatives unanimously, asserts that the two quantities are equal. Therefore, ~V2
(1/4)~2r 2
-
1
(10)
or
w=4. 46
THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1, 1995
(11)
So a n e w scientific law was declared by legislative fiat. The p r o p o n e n t s of this bill were so p r o u d of their discovery that they also decreed that [this] act introducing a new mathematical truth and offered as a contribution to education [is] to be used only by the State of Indiana free of cost by paying any royalties whatever on the same... In recent years, other m i d w e s t e r n legislatures have p r o n o u n c e d equally profound insights into subjects such as the origins of h u m a n k i n d and the m o m e n t w h e n life begins. One m o r e illustration from the physical sciences, from astronomy, m a y be amusing. Some 200 years ago, astronomers had discovered that the relative distances from the sun of the six planets then k n o w n could be expressed b y the series 4;
4+2o.3;
4+21.3;
4+22.3; ....
(12)
The first eight numbers of this series are 4, 7, 10, 16, 28, 52, 100, 196.
(13)
For agreement with the k n o w n distances of the planets, one must ignore the n u m b e r 28, or do as the astronomers led b y Bode did and postulate an undiscovered planet between Mars and Jupiter. They t h e r e u p o n proceeded to search for the missing planetary body. On the other hand, the great G e r m a n philosopher Hegel held the empirical approach in contempt [10]. Furthermore, he believed that the celestial bodies were ruled b y "immanent reason" and, hence, their distances must fit a "rational" sequence of numbers. This, he postulated, should be 1, 21, 31, 22, 32, 23~ 33
(14)
(except that he inserted 16 in place of 23). The interval between 22 and 32 was n o w large and hence in accord with that between Mars and Jupiter. Thus, there was no point in searching for a missing b o d y between these two planets. To Hegel's chagrin, the Bode group discovered Ceres within a year, and h u n d r e d s of small planets have since been located in that region. Hegel also expressed disparaging views of N e w t o n ' s s c i e n c e - - in this instance, the latter's celestial mechanics [111: The motion of the heavenly bodies is not a being pulled this way and that, as is imagined. They go along, as the ancients said, like blessed gods. The celestial conformity is not such a one as has the principle of rest or motion external to itself. It is not right to say, because a stone is inert, and the whole earth consists of stones, and the other heavenly bodies are of the same nature as the earth, therefore the heavenly bodies are inert. This conclusion makes the properties of the whole the same as those of the part. Impulse, pressure, resistance, friction, pulling, and the like, are valid only for other than celestial matter. Hegel had a talent for tailoring facts to fit his distorted theories.
N u m b e r Lore in the Life Sciences: Protein Studies When experimental data are relatively inaccurate, or sample only a small segi~ent of a field, individuals m a y be enticed into seeing simple integer relationships that provide a conceptual framework for arranging information. One example of perhaps excessive credulity appeared in the early days of analyses for amino acid composition of proteins. It is hard to appreciate today how difficult such analyses were before m o d e r n analytical techniques had been developed and automated. Nevertheless, heroic experimental efforts were made, and for a few proteins, values for the content of perhaps a dozen amino acids began to appear by the 1930s. Once some data had been accumulated, one of the most distinguished biochemists of the time, Max Bergmann, and his students began to search for some order in the amino acid compositions, and they soon found it [12]. Let me illustrate their approach and subsequent doctrine with a simple example first. For silk fibroin their analyses, normalized to a mole basis, gave the composition Glycine, 50 mole%, Alanine, 25 mole%. They then assumed that the first result could be interpreted to mean that glycine occurred periodically as every second amino acid in the sequence, that is, the sequence of residues in the protein chain of silk fibroin is G--X--G--X--G--X---G--X---G--X--G--,
(15)
where G represents glycine and X some other unspecified amino acid. Correspondingly, the alanine analysis was interpreted to signify the sequence A - - X - - X - - X - - A - - X - - X - - X - - A - - X - - X
. . . .
(16)
,
where A represents alanine. To make sequences (15) and (16) conformable, we shift the entries of the latter by one frame relative to the former and then fuse sequences (15) and (16) to obtain G--A---G--X---G--A--G--X--G--A---G--X . . . .
. (17) With subsequent analyses for lesser constituents, one could presumably fill in the unspecified positions X to obtain a complete sequence. Turning to a more complicated example, we can look at Table 2, which summarizes a report from Bergmann's laboratory of the composition of fibrin and lists the numerical relationships in amino acid content. Bergmann and N i e m a n n made the following assertion: Impressive stoichiometrical relationships have now been encountered in the case of two separate proteins, and it is obvious that these findings must have a role in the definition of the structure of the protein molecule. The simplest explanation lies in the assumption that the structural units [i.e., amino acid residues] of fibrin and gelatin are periodically arranged within the peptide chain and that ea4h unit exhibits its own particular periodicity. [For example, each glutamic acid residue in fibrin is separated from the previous, and the succeeding, glutamic residue, by seven other residues.] On the basis of this assumption the periodicities of the various units (amino acid residues) contained in fibrin were calculated and are recorded in [Table 2]. On examination of the periodicities of the various amino acids.., it was noted that all of the values can be considered as members of... arithmetical series [see the last column of Table 2]. . . . It is apparent that these expressions are derived from the prime numbers 2 and 3. This fact may be of significance in relation to the ability of these enormous protein molecules to assume a crystal structure. All of this, of course, ignored the very large experimental uncertainties in determinations of amino acid content. Furthermore, as Henry Bull, a very astute contemporary, pointed out [131, two different amino acids with a frequency of 3 and 7 along the peptide chain would collide with each other if they were invariant in periodicity:
Table 2. Amino Acids in Blood Fibrin Moles Amino Acid
Found
Calculated a
Ratio
Periodicity
Glutamic acid Lysine Arginine Aspartic acid Proline Tryptophane Histidine Methionine Cysteine
0.0959 0.0691 0.0442 0.0443 0.0443 0.0245 0.0161 0.0174 0.0124
0.0996 0.0664
72 48 32 32 32 18 12 12 9
8 12 18 18 18 32 48 48 64
0.0443 0.0443 0.0443 0.0249 0.0166 0.0166 0.0124
Key Integer 23 22 2 2 2 25 24 24 26
x x x x x x x x x
3o 3 32 32 32 3 3 3 3o
a Taking a base value of 0.0443 for average of moles of arginine, aspartic acid, and proline. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
47
Separate Glycine (1 : 3)
G--X--X---G--X--X---G--X--X---G--X--X--G--X--X--G . . . .
Proline (1 : 7)
Pr--X--X--X--X--X--X--Pr--X--X--X--X--X--X--Pr--X ....
(18~ .
(19)
Combined (with frame shift) G--Pr--X---G--X--X---G---X--Pr---G--X--X---G--X--X-- G r --X . . . .
(20)
OF
G--X--Pr--G--X--X---G--X--X-- P G r --X--X---G--X--X---G--Pr . . . .
Analyses of gelatin had given frequencies of I in 3 and 1 in 7 for glycine and proline, respectively. Having "established" their principles of protein structure, Bergmann and Niemann found further confirmation in published analytical data for hemoglobins, orosins, and keratins. Bergmann thereupon extended the implications of his stoichiometric relationships and reached the conclusion that all proteins are constituted of 288 residues or an integer multiple thereof. As the Bergmann theory requires that the number of residues of each constituent amino acid must be exactly divisible into the total number, the value of 288 for the latter is, as H. Bull [13] noted, a particularly felicitous one, for 288 has the largest number of integer divisors of any number from 0 to 576. Thus, the analytical composition data need the least rounding off to fit one of the acceptable divisors. In protein biochemistry, the number "288" soon attained a prominence similar to that of "137" in astrophysics. It began to pop up everywhere. A contemporary of Bergmann's, D. M. Wrinch, a respected mathematician acquainted with X-ray crystallography, proposed at this time a very novel structure for the fundamental structural unit of proteins [14]. It may be visualized as a tautomeric form of a diketopiperazine (a known structure for two coupled amino acids),
H /
\N-H
H--N
\ o//c
c
/
t ~R H
(Diketopiperazine)
H R ~CJ
I
HO ~ C
C/O H
\N--
C
/
/
t ~R H (Wrinch cyclol)
in which the open atom bonds lead to fusion of the hexagonal cyclol rings to generate a macromolecular structure. Wrinch predicted that a particularly stable multicyclol would be one constituted of 288 amino acid residues, a number in exact concordance with Bergmann's. 48
THE MATHEMATICALINTELLIGENCER VOL. 17, NO. 1,1995
(21)
Also contemporary with the Bergmann theory was the development of the modern analytical ultracentrifuge by Svedberg and his co-workers. This tool for the first time allowed one to determine masses of proteins of widely different molecular weight, from 104 to 107. By 1939, the molecular weights of some 50 proteins had been determined [15]. That these biomacromolecules might be constituted of multiple subunits was a tempting concept. Indeed, Svedberg became convinced early that proteins fell into classes of integral multiples of a fundamental structural unit of molecular weight 17,600, the integers being in the sequence 2, 4, 6, 8, 16, 24, 48, 96, 192, 384 (which can also be expressed in the form 2n3m). A protein of 2 x 17,600 molecular weight, given the usual average amino acid residue weight, would be constituted of 288 amino acids. So he opined, Nature in the production of organic substance within the living cell seems to work only with a very limited number of main lines .... The numerous proteins are built up according to some general plan which secures for them only a very limited number of different molecular masses and sizes .... Again the inveterate iconoclast H. Bull [13] pointed to the substantial experimental uncertainties in the values of the molecular weights, which made precise, large integer divisors, such as 96, 192, 384, suspect. As he said, "all numbers, if they are large enough, are approximate multiples" of some smaller unit. Bull also surmised that the clustering around 17,000, and its multiples, merely reflected the limited data at that time, as, indeed, has turned out to be true. Although most people think that the concept of a protein helix was revealed to Linus Pauling on Mount Pasadena in 1950, it actually had a tortuous earlier history. The slow evolution of the helix, or the spiral as it was originally called, as a central structural element in proteins also reflects the influential grip of integers. The most incisive early analysis of protein structure problems came from M. L. Huggins [16]. In 1943, in a publication in Chemical Reviews based on a talk he had given at an American Chemical Society symposium in 1937, he explicitly formulated the necessary constraints of bond distances
and angles, as well as the requirement for N - - H . . . O hydrogen "bridges." Then he showed that spirals with two residues or three residues per turn could account for the o 5.1-A X-ray reflection characteristic of c~-keratin. Furthermore, he pointed out that a threefold screw axis of symmetry would give a translation along the axis of the spiral of about 1.7 A. Indeed, a meridional reflexion near 1.5 K was discovered at that time by I. MacArthur [17], but this was ignored or overlooked by protein structure scientists. In a prophetic sentence, Huggins also noted "that there is nothing about this [spiral] structure which requires exactly three residues per turn." In fact, he felt from the models he had made "that the bond distance and angle requirements are best satisfied by a slightly smaller [i.e., noninteger] number of residues per turn." Pathetically, like Moses, Huggins got a glimpse of the Holy Land but never entered it. Ultimately, Pauling and Corey [18] broke loose unequivocally from integer constraints and showed, originally by model-building, that the nonintegral 3.7-residue turn best fitted interatomic structural and energetic requirements. Thus was born the c~-helix of proteins. It is also instructive to look back at some of the meanderings of hemoglobin research in the mine fields of numerology. When the first accurate iron content of hemoglobin (0.335% Fe) was determined a century ago, it became possible to calculate a (minimal) molecular weight, which turned out to be 16,700. Not surprisingly, experimental osmotic pressure measurements at that time gave 17,000. Hence, an equation for uptake of oxygen (02) by hemoglobin (Hb) could be written in the form of a simple chemical equation, Hb + 02 = HbO2 (22) with a 1 : 1 molecular ratio for the reactants. When the participants in a chemical transformation are in equilibrium, thermodynamicists can show that the first two laws of thermodynamics demand that the ratio of concentrations of the species must be a constant, denominated an equilibrium constant, K. For the uptake of 02 by Hb, this would be expressed as (HbO2)
(Hb)(O2)
- K,
g(o2)
(24)
1 + K(O2)
The graphical equivalent of Eq. (24) is a rectangular hyperbola. Indeed, when G. Hfifner measured fractional saturation as a function of oxygen pressure [19] and graphed the results, they fitted a hyperbola (Fig. 1, left curve).
/ J
f
/"
/
.~" 7 f
j
//
f
/ /
// f j' / /
2O
40
~O
Bo
Ioo
Figure 1. Oxygen uptake curves for hemoglobin. Ordinate, percent saturation of hemoglobin; abscissa, pressure of oxygen gas (in mm.). The left, hyperbolic curve is that reported by G. Hiifner (19), the right, sigmoidal curv6 that obtained by C. Bohr (20).
Subsequently, Christian Bohr (father of Niels Bohr) carried out one of the classic experimental studies in the history of science, carefully controlled measurements of the uptake of 02 by hemoglobin [20]. His investigations established that the uptake curve was certainly not a hyperbola but showed instead an S-shaped curvature (Fig. 1, right curve). It is impossible for a unisite hemoglobin, that is, one capable of binding only a single 02 molecule, to manifest an S-shaped oxygen-uptake curve. The protein, therefore, must have a molecular weight that is a multiple of 17,000, that is, it must have more than one iron atom, each of which can hold one 02. Not surprisingly, other osmotic pressure measurements gave values higher than 17,000, but the results reported were not concordant. The famous biophysicist A. V. Hill then proposed [21] the following chemical equation for 02 uptake by a multisite hemoglobin, represented as Hb n (n an integer >1): Hb n + nO2 = Hbn(O2)n.
(23)
where the parentheses indicate the equilibrium concentration of the enclosed species. Simple algebraic analysis then led to an equation for the fractional saturation of this oxygen carrier: Molecules oxygen-filled Hb Molecules empty Hb + molecules filled Hb _
I
(25)
This is a simplistic formulation (as Hill himself recognized), but at the time, it was a giant leap forward. Such a chemical presentation leads to the following logarithmic equation for 02 uptake: log
[oxygenated Hb] [ n ~ H b ] J
= log K + n log(O2). (26)
From Eq. (26) one would expect that a graph of log {[oxyl/[nonoxy]} versus the logarithm of pressure (or concentration) of 02 should be linear and have a slope of n, the number of binding sites for oxygen in hemoglobin. For decades, published experimental oxygenation data did, indeed, always fall on a line in a log-log graph, But a new problem was encountered: n values observed were THE MATHEMATICALINTELLIGENCER VOL. 17, NO. 1, 1995
49
not an integer, but clearly a decimal number, around 2.8. But it was assumed [and later proven] that all hemoglobin molecules are alike (Fig. 2) and each has exactly four iron atoms. Thus, a new enigma succeeded the previous one. Ultimately, G. S. Adair's precise osmotic pressure measurements [22] established that hemoglobin is a tetramer of heme(iron)-containing subunits, which was soon confirmed by Svedberg's early ultracentrifugation studies. Thereupon, the stage was set for Adair's expression of the oxygen equilibria of hemoglobin in terms of four stoichiometric chemical steps: HD q- 0 2 : HbO2, HDO2 -+- O2 : Hb(O2)2~ HD(O2)2 q- O2 ~-~HD(O2)3, HB(O2)3 q- 0 2 = HD(O2)4
(27)
with four associated equilibrium constants. Thus, the smooth continuous character of the oxygen-uptake curve for hemoglobin was made consonant with sharp stoichiometric steps by assuming there are four such steps in concert. It has been shown recently [23] that all oxygen-binding curves for hemoglobin can be fully described by an algebraic expression containing four (complex) binding constants, namely, Moles bound 0 2 Total moles hemoglobin
0.30e 0"24~i(02)
1 + 0.30e~ +
0.30e-0.241ri (O2)
1 + 0.30r
0.17e0.70~i(O2) 1 + 0.17e0'70"i(O2) 0.17e-0"70~i(O2) + 1 + 0.17e-~ +
(28) Many protein scientists are made uneasy by these "imaginary" numbers in an equation leading to a "real" experimentally measurable quantity [the left-hand side of Eq. (28)]. The same individuals are not at all distressed by their use of "irrational" numbers in equations similar to Eq. (28) describing fully rational experimental measurements. Once again, we see how the transfer of scientific terms with a long ancestry in common parlance can lead to gross misconceptions. This review has focused largely on the Lorelei influence of integers, luring some scientists onto destructive shoals. It would be wrong, however, to create the impression that an attraction to integers is always perilous or misleading. On the contrar~ postulations of integer relationships have played pivotal roles in the construction of powerful theories in the sciences. One need only cite quantum numbers in theoretical physics, integer stoichiometries in atomic chemistry, and Mendelian integers 50 THEMATHEMATICALINTELLIGENCERVOL.17,NO.1, 1995
Figure 2. Side v i e w of scale molecular model of human h e m o g l o b i n (molecular weight about 64,000). The actual model is about 60 cm. in diameter. All of the thousands of atoms except hydrogens are s h o w n individually in the model. The scale is I cm. -- 1 A. At the center of the valley shown at the top is a two-fold symmetry axis, on the edge of a tetrahedron of the four constituent subunits (each of 16,000 molecular weight). Scale models of this type are invaluable in efforts to relate biological function to molecular structure.
in genetics. Let me elaborate with the chemical example, with which I am most familiar. John Dalton, the formulator of modern atomic theory, was struck by the observation that when carbon, C, reacts with oxygen, O, two different compounds can be formed; the ratio of oxygen content to carbon content is twice as great for one as for the other. From this fact, and a very few others, he formulated the universal principle known as the "law of multiple proportions": When two elements form a series of compounds, the ratios of the masses of the second element that combine with a fixed mass of the first element can always be expressed as small integers. This law is the foundation upon which all of structural atomic chemistry has flourished for the past two centuries. Fortunately, not very much chemistry was known in Dalton's time. By the end of the 19th century, it was apparent that Dalton's law is not strictly true; stoichiometries (proportions of elements, or atoms, in a compound) are not always expressible as small integers. A century ago, a class of substances now known as clathrate hydrates was discovered in which the measured stoichiometries give numbers such as M. 72H20 (where M is any one of a large series of atoms or molecules)--hardly "a small integer" relationship. In very recent times, the new, very exciting superconductors have been found to show a continuum of stoichiometries; for example, the perovskites have the formula YBa2Cu3Ox, where x is variable (average value near 6.527), depending on the sample prepared. Also, as described in this review, there is really a continuum of compounds of hemoglobin with oxygen, with stoichiometries for H b . (O2)x ranging from x=Otox =4.
Yet these seeming contradictions have not invalidated Dalton's law. Atomic theory has been so successful and has become so deeply ingrained in chemical thinking that it has been assumed that experimental negations are only apparently so and will be explained in time. I have shown in this article h o w an acceptable rationalization was achieved, over a period of decades, for the hemoglobinoxygen system. Resolution of the c o n u n d r u m s presented by the clathrate hydrates was achieved in more recent decades as X-ray diffraction revealed the very fine details in the complex structures of large molecules. Thus, M . 72 H20 is really a m u c h larger molecule that should be described as M6046H92. Integer stoichiometry still exists, although the pertinent integers might not be considered small. In the perovskites, X-ray diffraction shows that the atoms are arranged in a megastructure in infinite planes, so that the entire crystal constitutes a "molecule" of the substance even w h e n occasional positions therein are unoccupied by an oxygen atom. Thus, in view of the history of chemical integer stoichiometry, it was reasonable for Bergmann to ass u m e that the amino acids, glycine, alanine, a n d so on, should be present in protein molecules in quantities related to each other by integers. The contradiction pointed out by Bull, that frequencies of 3 and 7 found in gelatin are incompatible, could be set aside as an enigma to be resolved in the future. Similarly, Svedberg's attempt to fit all protein molecular weights into a limited number of classes with integral multiples of a fundamental unit weight was concordant with central modes of thinking in structural chemistry. Ironically, although the specific form of Svedberg's proposal did not withstand the subsequent waves of precise data for molecular weights, a less restrictive visualization--that large proteins are constituted of smaller identical subunits--has proved valid, and has been very useful in revealing unsuspected point symmetries in the structural arrangements of these supermolecules. Can one discern any features that clearly distinguish the successes from the failures in applications of "number mysticism"? There are some examples of failures where ignorance or stupidity underlies the proposal-attempts to require ~r to be an integer, Procrustean distortions such as those of Mills to twist composites of fundamental constants to fit into the same bed (the integer 3). Others are founded on revelation, often religious but including ideas such as "immanent r e a s o n ' - - h a r d l y accessible to scientific scrutiny. The tantalizing enigmas are the constructs based on numbers such as 137 and 288. With luck, Eddington, Bergmann, Wrinch, and Svedberg might all have been right. Thus, it is difficult to k n o w a priori w h e n integers will give constructive or misleading insights. There is nothing intrinsically wrong with number mysticism. This situation reminds me of an a n o n y m o u s poetic supplication I discovered some' years ago, which provides a fitting conclusion for this article [24]:
Grant, oh God, thy benedictions On my theory's predictions, Lest the facts when verified, Show Thy servant to have lied.
References 1. T. Dantzig, Number, First Edition, New York: Macmillan (1930); Fourth Edition, Garden City, NY: Doubleday and Co. (1954). 2. P. J. Davis and R. Hersh, The Mathematical Experience, Birkh/iuser, Boston (1981). 3. J.B. Kaper and H. L. T. Mobley, Immortal sequence, Science 253 (1991), 951-952; 254 (1991), 358. 4. D.J. Gross, Can we scale the Planck scale?, Physics Today 42, No. 6 (June 1989), 9-11. 5. S. "Chandrasekhar, Eddington, The Most Distinguished Astrophysicist of His Time, Cambridge: Cambridge University Press (1984). 6. G. Beck, H. Bethe, and W. Riezler, Bemerkung zur Quantentheorie der Nullpunktstemperature, Naturwissenschaflen 2 (1931), 39. 7. J. E. Mills, Relations between fundamental physical constants, J. Phys. Chem. 36 (1932), 1089-1107. 8. P. Stanbury, The alleged ubiquity of 7r, Nature 304 (1983), 11. 9. P. Beckman, A History of Tr(Pi). 2nd ed., Boulder, CO: Golem Press (1972). 10. M. Polanyi, Passion and controversy in science, Lancet (1965), 921-925. 1I. E G. Tait, Thermodynamics, Edinburgh: David Douglas (1877), 57. 12. M. Bergmann and C. Niemann, On blood fibrin, a contribution to the problem of protein structure, J. Biol. Chem. 115 (1936), 77-85. 13. H. B. Bull, Protein structure, Adv. Enzymol. 1 (1941), 1-42. 14. D. Wrinch, Is there a protein fabric? Cold Spring Harbor Symposia on Quantitative Biology, 6 (1938), 122-134. 15. T. Svedberg, A discussion of the protein molecule, Proc. Roy. Soc. (London) B127 (1939), 1-17. 16. M. L. Huggins, The structure of fibrous proteins, Chem. Rev. 32 (1943), 195-218. 17. I. MacArthur, Structure of a-keratin, Nature 152 (1943), 3841. 18. L. Pauling and R. B. Corey, Atomic coordinates and structure factors for two helical configurations of polypeptide chains, Proc. Natl. Acad. Sci. (USA) 37 (1951), 235-240. 19. G. Hfifner, Neue Versuche fiber die Dissociation des Oxyh/imoglobins, Arch. Anat. Abt. 5 (1901), 187-217. 20. C. Bohr, Theoretische Behandlung der quantitative Verh/iltnisse bei der Sauerstoffaufnahme des H/imoglobins, Zentralbl. Physiol. 17 (1903), 682-688. 21. A. V. Hill, The possible effects of aggregation of the molecules of hemoglobin on its dissociation curve, J. Physiol. (London) 40 (1910), iv-vii. 22. G. S. Adair, The osmotic pressure of hemoglobin in the absence of salts, Proc. Roy. Soc. (London) A109 (1925), 292300. 23. I. M. Klotz, A perspective into ligand-receptor affinities using complex numbers, Proc. Natl. Acad. Sci. (USA) 90 (1993), 7191-7194. 24. Anonymous, European ScientificNotes (U.S. Office of Naval Research, London), February 20, 1963, page 20. Department of Chemistry Northwestern University Evanston, IL 60208-3113, USA THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
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II
I
Four Encounters w i t h Sierpifiski's G a s k e t 1 Ian Stewart
Mathematicians would not be happy merely with simple, lusty configurations. Beyond these their curiosity extends to psychopathic patients, each of whom has an individual case history resembling no other; these are the pathological curves of mathematics. Edward Kasner and James Newman
Mathematics and the Imagination
year Sierpifiski gave the first systematic lecture course ever taught on set theory. H e published a book based on it in 1912, which was a m o n g the first texts on that subject. Sierpifiski had found his subject, and the bulk of his subsequent research was in set theory and point set topology.
One of the most fascinating features of mathematics is the w a y in which the same idea crops u p again and again in apparently unrelated areas. Over the last few years, I have been haunted b y the object that Benoit Mandelbrot [1] has christened SierpMski's gasket. It is the triangular fractal s h o w n in Figure 1. N o w a d a y s , fractals are respectable, and the sentiments expressed in the above quotation seem old-fashioned: it shows h o w m u c h attitudes have changed. Most people's, anyway. Sierpifiski's gasket arises naturally in m a n y branches of mathematics, and m y aim is to convince you that without it, mathematics w o u l d be the poorer. But first, a few w o r d s about the m a n himselL
Explorer of the Infinite W a d a w Sierpi~ski was born in Warsaw, Poland, on 14 March 1882. His father Konstanty Sierpifiski was a doctor. M y source, Kasimierz Kuratowski [2], Vol. 1, fails to record any details about his mother. Sierpi~ski studied u n d e r the n u m b e r theorist G. Voronoi, and his early w o r k was also in n u m b e r theory, a topic to which he repeatedly returned in later life. He obtained a doctorate in 1906, and by 1909 he h a d m o v e d to the University Jean Casimir in Lvov, becoming a professor there in 1910. The year 1909 is more significant, however, because in that
1This article is an expanded and somewhat rewritten version of the Lonseth Lecturegiven at Oregon State University,Corvallis on 14 May 1991, and the London Mathematical Society Popular Lecture given at Sheffield University on 17 June 1991 and Imperial College London on 28 June 1991. 52
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1 (~1995 Springer-Verlag New York
Figure 1. The SierpiIiski gasket. During the First World War, Sierpi~ski was interned in Russia, first at Viatka and then in Moscow. There he worked with the Russian mathematician N. Lusin on projective sets and real functions. Their first joint paper appeared in 1917 and the last in 1929. At the end of the war, Sierpir~ski returned to Lvov, but almost immediately became a professor at the now reestablished University of Warsaw. The place was a hotbed of Polish mathematics, specialising in set theory and foundational matters, with people such as Zygmunt Janiszewski, Stefan Mazurkiewicz, and Jan Lukasiewicz. Together with Sierpifiski, the first two started their own journal, Fundamenta Mathematicae, which exists to this day. Lvov, too, became a major centre of Polish mathematics under Stefan Banach--see Ciesielski [3, 4] for the atmosphere of this period, including the story of the famous "Scottish Caf4" in Lvov. Between the wars, Sierpifiski's talents flourished. He was always prolific: his collected works include 720 papers published between 1906 and 1968, 106 expository articles, 50 books (plus 7 at the level of secondary education), and 12 mimeographed sets of lecture notes. The start of World War II found him still in Warsaw, where he continued his scientific work as best he could, teaching
clandestine courses at the university to small audiences. After the uprising of 1944, he was deported by the Germans to the region around Krak6w. In 1945, he briefly lectured at the Jagiellonian University of Krak6w, before returning once more to Warsaw. In 1958, he wrote a major monograph, Cardinal and Ordinal Numbers. He remained very active if~ administrative matters and received a number of important prizes and other honours from the Polish government. He died in Warsaw on 21 October 1969. Following Sierpifiski's wishes, his grave bears just two words (in Polish): Explorer of the Infinite.
Encounter 1: Sierpil~ski's Encounter with Sierpiliski's Gasket (Wadaw Sierpifiski, 1915) My claim is that (well before Koch,Peano, and Sierpi~ski)the tower of Gustav Eiffelbuilt in Paris deliberately incorporates the idea of a fractal curve full of branch points. Benoit Mandelbrot
The Fractal Geometry of Nature The gasket made its first appearance in an article only three and a half pages long [5]. (Though, being published in Comptes Rendus, it couldn't have been very THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
53
m u c h longer without infringing the rule brought i n - - i t is s a i d - - to prevent Augustin-Louis Cauchy from filling every issue with vast screeds.) A more detailed treatment followed a year later [6]; see also Ref. 2. The gasket's role was to provide an example of "a curve simultaneously Cantorian and Jordanian, of which every point is a point of ramification"; less formally, a curve that crosses itself at every point. A point of ramification of a curve C is a point p such that there exist three subsets of C, all continua, of which any pair intersect only at p. Sierpifiski's o w n diagram of the construction of this curve is shown in Figure 2(a). He first establishes that it is a Cantor curve (a continuum that is not dense in the plane). By a careful study of the process by which the various triangles and subtriangles are constructed, he then proves that every point other than the three vertices of the original triangle is a point of ramification. These three vertices are clearly not points of ramification, but
(a)
before dealing with them, Sierpifiski offers Figure 2(b) as a sketch proof that his set is also a Jordan curve. Finally, he observes that if six copies of his triangle are arranged to form a regular hexagon, then the result is a Cantor and Jordan curve for which every point is a point of ramification. Sierpifiski's curve is of course a fractal, though that word was not coined until 1975 by Mandelbrot [1], w h o also, in jest, introduced the term Sierp#fski gasket. At about the same time, Sierpifiski invented several other celebrated fractals, including his space-filling curve [7] and the Sierpifiski carpet [8]. He also invented several functions with fractal properties: a function [9] that has zero derivative almost everywhere, yet climbs monotonically from 0 to 1 (a forerunner of the "devil's staircase" [1]), and a function f such that f ( f ( x ) ) = x whose graph is dense in the plane [10]. (You might like to try to construct such a function; see below for Sierpifiski's solution, a typical example of his ingenuity.) Because the gasket is assembled from three copies, each half the size, its fractal (or Hausdorff-Besicovitch) dimension is log 3 / l o g 2 = 1.5849 . . . . See Ref. 11 for details. It has a three-dimensional relative, to which Mandelbrot [1] gives the less inspired name "a fractal skewed web" (Fig. 3), but which I prefer to call the Sierpi~ski cheese. Curiously, this has fractal dimension log 4/log 2 = 2, the same as that of an ordinary Euclidean plane. Observe that the section cut away at each stage is not an inverted tetrahedron, which is w h y tetrahedra - - contrary to Arist o t l e - do not tile space.
Solution
To get a function f such that f ( f ( x ) ) = x with a dense graph, define f ( a + b y e ) = b + a v ~ for rational a and b; otherwise define f ( x ) = x. Sierpifiski invented a fractal 60 years before the word existed. Mandelbrot - - with some justification-suggests that Eiffel invented the moral equivalent of the Sierpifiski gasket 26 years before Sierpifiski did. A year later, in 1890, another Frenchman characterised a combinatorial incarnation of the Sierpifiski gasket:
Encounter 2: Pascal's Encounter with Sierpifiski's Gasket (Edouard Lucas, 1890)
I have yet to see a problem, however complicated, which when you looked at it in the right way, did not become still more complicated. Poul Anderson
Figure 2. Sierpifiski's version. 54 THEMATHEMATICAL INTELLIGENCERVOL.17,NO.1,1995
As the section title shows, attributions for this encounter are tricky: The material goes back so far into the collective mathematical consciousness that it is difficult to award credit to any specific person. Instead, we record
Figure 3. The Sierpifiski cheese.
some milestones. Pascal gets credit for the encounter because it is his name that is attached to the triangular array of binomial coefficients (~). Like most attributions from the distant past (and many from the near present), it is utterly w r o n g - - for example, the triangle appears on the title page of an early 16th-century arithmetic by Petrus Apianus; it can be found in a Chinese mathematics book of 1303; and, indeed, it has been traced back at least to Omar Khayy~m around 1100, who almost certainly got it from earlier Arabic or Chinese sources. Michael Stifel introduced the term binomial coefficient around 1500. The explicit formula n ! / r ! ( n - r)! was given by Isaac Newton and permitted the nonrecursive computation of binomial coefficients. In its interpretation as the number of ways to choose r items from a set of n, this expression (though not in that notation) was known to Bhaskara (b. 1114). What is the parity of (~)? That is, what is its value (mod 2)? It is easy to experiment with a computer because it suffices to implement the rule of formation of the triangle mod 2. The result, Figure 4, is striking and surprising.
Figure 4. Pascal mod 2. THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1,1995 5 5
The odd binomial coefficients form a discrete variant of the Sierpifiski gasket. Indeed, b y suitably rescaling segments of the figure and taking an appropriate limit, we m a y consider the parity-coloured Pascal triangle to be a second manifestation of the gasket. The ultimate explanation for this pattern is a theorem attributed b y Dickson [12] to Edouard Lucas in 1890:
It follows that almost all binomial coefficients are even. Singmaster [18] takes this observation further, proving that for a n y m, almost all binomial coefficients are divisible by m. A refinement of Lucas's t h e o r e m was p r o v e d b y Glaisher [19]. Similar results w e r e k n o w n to K u m m e r [20].
T H E O R E M 1: Let p be a prime. Write n and r in p-ary notation: n = nk . . . no, r = rk " "to, where the n d and rj are 0, 1 , . . . , p - 1. Then
T H E O R E M 2: Let p be prime. The largest power of p that divides (~) is equal to the number of carries in the (mod p) addition of r and n - r.
(n)_ h (rn~) (modp).
A proof is given in Ref. 21. Related articles include Refs. 22-26.
j=0 We make the standard convention that if r does in the range 0 <_ r < n, then the value of (~) is attractive convention which differs from this if n negative m a y be f o u n d in Ref. 13".) Taking p = observing that (~)=
(10)= (11)=1 ,
not lie 0. (An or r is 2 and
(~)=0,
it follows that (~) is even if and only if some binary digit of n is 0 while the corresponding digit of r is 1. Similar statements can be m a d e for any prime p. Lucas's theorem explains the recursive Sierpklski pattern of Figure 4. It implies that w h e n n and v are expressed in binary as above, the parities of
(n)
/'k
" "ro
(2k+:+n)=llnk'"no) \ Ork
2k+l+n) 2k + l +
ro
'
= ~lnk'"no) \lrk
In the great temple at Benares, beneath the dome which marks the centre of the world, rests a brass plate in which are fixed three diamond needles, each a cubit high and as thick as the body of a bee. On one of these needles, at the creation, God placed sixty-four discs of pure gold, the largest disc resting on the brass plate, and the others getting smaller and smaller up to the top one. This is the Tower of Bramah. Day and night unceasingly the priests transfer the discs from one diamond needle to another according to the fixed and immutable laws of Bramah, which require that the priest on duty must not move more than one disc at a time and that he must place this disc on a needle so that there is no smaller disc below it. When the sixty-four discs shall have been thus transferred from the needle on which at the creation God placed them to one of the other needles, tower, temple, and Brahmins alike will crumble into dust, and with a thunderclap the world will vanish. M. de Parville La Nature, 1884
..to
are the same, whereas n = (2k+1 + r ) I Oak''" is always even. The patterns for (~) (mod k), some of which are shown in Figure 5, are at least as pretty as that for k = 2. They have been discussed in this journal by Sved [141; see also Refs. 15 and 16. W h e n k = p~ is a p r i m e power, there is a generalization of Lucas's theorem, d u e to Kazandzidis [171, which explains the patterns in the same m a n n e r as Lucas's theorem does and relates them to the base k expansions of n and r. For other values of k, it seems to be necessary to write k as a p r o d u c t of prime powers kj and consider expansions to all bases k~. As the gasket is a fractal of dimension less than 2, its area (two-dimensional Hausdorff measure) is zero. 56
Encounter3: Hinz's Encounterwith Sierpifiski's Gasket (AndreasHinz, 1990)
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
E d o u a r d Lucas also seems to have been h a u n t e d - albeit u n w i t t i n g l y - - b y Sierpifiski's gasket. In 1883, he marketed the puzzle k n o w n as the Tower of Hanoi, u n d e r the p s e u d o n y m N. Claus [27]. It is similar to de Parville's romantic Tower of Brahma but uses eight (or fewer) discs m a d e of m o r e m u n d a n e materials; moreover, instead of needles, it employs pins. It is an old friend of recreational mathematicians and computer scientists. N u m b e r the pins 1, 2, and 3: A s s u m e all discs start on pin I and m u s t end on pin 2. It is well k n o w n that the solution has a recursive structure, explained, for example, in Ref. 28. I call it the army method. This particular a r m y has a large n u m b e r of privates, w h o have been trained to solve 1-disc Hanoi but nothing more ambitious: "Move the disc to pin 2 while keeping it in order of s i z e - - y e s Bloggs, I k n o w y o u can't change the order of one disc, but I w o u l d n ' t be surprised if y o u 'orrible lot f o u n d a way to screw even that up."
Figure 5. Pascal mod k. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
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The corporals can solve 2-disc Hanoi: "Private Bloggs: Move the smaller disc to pin 3 as in 1-disc Hanoi. Private Aggs: move the larger disc to pin 2. Private Bloggs: Move the smaller disc to pin 2 as in 1-disc Hanoi." The sergeants can solve 3-disc Hanoi: 9 "Corporal Cloggs: Move the top two discs to pin 3 as in 2-disc Hanoi. Private Aggs: Move the larger disc to pin 2. Corporal Cloggs: Move the top two discs to pin 2 as in 2-disc Hanoi." The lieutenants can solve four-disc Hanoi: 9 "Sergeant Doggs: Move the top three discs to pin 3 as in 3-disc Hanoi. Private Aggs: Move the larger disc to pin 2. Sergeant Doggs: Move the top three discs to pin 2 as in 3-disc Hanoi." Private Aggs, of course, is a specialist. You should be able to write down how Captain Eggs, Major Foggs, Colonel Goggs, and n-star Generals Hoggs, and so forth, solve their own levels of the puzzle. To solve n-disc Hanoi for n > 8 takes an (n - 7)-star General. I once tried to convince Yorkshire Television to enact this method for 5-disc Hanoi, using real soldiers. Unfortunately, they decided it would put undue strain on the audience's attention span. That solves the puzzle: What else is there to say? Rather a lot, actually. You can pose many more subtle probl e m s - such as how to move from any given position to any other in the most efficient m a n n e r - - n o n e of which are within the capabilities of my monstrous regiment.
Sometimes it helps to think geometrically. With any puzzle of this general type (moving objects, finite number of positions) we can associate a graph. Its nodes are the positions, its edges the moves between them. The graph Hn for n-disc Hanoi was introduced by Scorer et al. [29], rediscovered by Er [30], re-rediscovered by Lu [31], and (re)Bdiscovered by me [15]. It is in the nature of the topic that most people working on it don't know the literature, so (re)ndiscovery for all n > 4 will occur with probability I in the long run. What does Hn look like? For definiteness, consider H3, which describes the positions and moves in three-disc Hanoi. To represent a position, number the three discs as 1, 2, and 3, with 1 being the smallest and 3 the largest. Number the pins 1, 2, and 3 from left to right. Suppose, for example, that disc 1 is on pin 2, disc 2 on pin 1, and disc 3 on pin 2. Then we have completely determined the position, because the rules imply that disc 3 must be underneath disc 1. Thus, we can encode this information in the sequence 212, the three digits, in turn, representing the pins for discs 1, 2, and 3. Therefore, each position in 3-disc Hanoi corresponds to a sequence of three digits, each being 1, 2, or 3. There are 33 = 27 positions (because each disc can be on any pin, independently of the others). What are the permitted moves? The smallest disc on a given pin must be at the top, so it corresponds to the first appearance of the number of that pin in the sequence. If we move that disc, we must move it to the top of the pile on some other pin, that is, we must change the number so that it becomes the first appearance of some other number. For example, in position 212, suppose we wish to move disc 1. This is on pin 2 and corresponds to the first occurrence of 2 in the sequence. Suppose we change this first 2 to 1. Then this is (trivially!) the first occurrence
212
112 Figure 6. Moves in 3-disc Hanoi from position 212. 58
THE MATHEMATICALINTELLIGENCERVOL, 17, NO, 1,1995
312
232
of the digit 1; so the move from 212 to 112 is legal; so is 212 to 312 because the first occurrence of 3 is in the first place in the sequence. We may also move disc 2, because the first occurrence of the symbol I is in the second place in the sequence. But we ~annot change it to 2, because 2 already appears earlier, in the first place. A change to 3 is, however, legal. So we may change 212 to 232 (but not to 222). Finally, disc 3 cannot be moved, because the third digit in the sequence is a 2, and this is not the first occurrence of that digit. To sumup: From position 212 we can make legal moves to 112, 312, and 232, and only these (Fig. 6). Proceeding in this way, we list all 27 positions and all possible moves by following the above rules. The graph /-/3 can then be constructed: The result (after some rearrangement for elegance) is Figure 7. Something that pretty can't be coincidence! //3 consists of three copies of a smaller graph, linked by three single edges to form a triangle. But each smaller graph, in turn, has a similar triple structure. Why does everything appear in threes, and why are the pieces linked in this manner? In fact, the graph/-/2 looks exactly like the top third of Figure 7. Even the labels on the vertices are the same, except that the final I is deleted. It is, of course, easy to see this without working out the graph again. You can play 2-disc Hanoi with three discs: just ignore disc 3. Suppose disc 3 stays on pin 1. Then we are playing 3-disc Hanoi but restricting attention to those three-digit sequences that end in 1, such as 131 or 221. These are precisely the sequences in the top third of the figure. Similarly, 3disc Hanoi with disc 3 fixed on pin 2 corresponds to the lower left third, and 3-disc Hanoi with disc 3 fixed on pin 3 corresponds to the lower right third. It works for the same reason that the army method does. This explains why we see three copies of the 2-disc Hanoi graph in the 3-disc graph. A little further thought shows that these three subgraphs are joined by just three single edges in the full puzzle. For, in order to join up the subgraphs, we must move disc 3. When can we do this? Only when one pin is empty, one contains disc 3, and the other contains all the rest! Then we can move disc 3 to the empty pin, creating an empty pin where it came from and leaving the other discs untouched. There are six such positions, and the possible moves join them in pairs. The same argument works for any number of discs, so Hn+l consists of three copies of Hn linked at the corners. For example, Figure 8 shows/-/5. As the number of discs becomes larger and larger, the graph looks more and more like the Sierpifiski gasket. You can use the graph to answer all sorts of questions about the puzzle. For example, it follows inductively that the graph is connected, so you can always move from any position to any other. The minimum path from the usual starting position to the usual finishing position runs straight along one edge of the graph, so (again by
J
Figure 7. The graph H s of 3-disc Hanoi.
G G .... G
GGGG Figure 8. The graph H 5 of 5-disc Hanoi.
induction) has length 2 n - 1. This result, long assumed in the form "the largest disc moves only once," was first proved by Wood [32]. The resemblance of H,~ to the Sierpifiski gasket has a curious application. Not long after Ref. 15 appeared, I attended the International Congress of Mathematicians in Kyoto, and a German mathematician named Andreas Hinz introduced himself. He had been trying to calculate the average distance between two points in a Sierpifiski gasket of unit side, encountered difficulties, and asked two experts. Here's what they said. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
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The Considered View of Expert 1: It's very difficult. The Considered View of Expert 2: It's trivial, and the answer is 8/15. Here is Expert 2's proof. The idea is first to find the average distance a to some particular corner, and then to use that to find the average distance d between two arbitrary points. From Figure 9, it follows immediately that a=~ 112 +2 /2 +
~)] ,
Figure 9. Proof that (x = 2
Figure 10. Proof (?) that d = 8/15. 60
THE MATHEMATICAL INTELLIGF_.NCERVOL. 17, NO. 1, 1995
from which a = 2. Now consider two points: They are either in the same subtriangle, as in Figure 10(a), or not, as in Figure 10(b). The respective probabilities are 89and 2. In the latter case, the shortest path between them goes through the common vertex. Therefore, d=~+~
2(2) 2 ,
from which d = 8/15. Happy? You shouldn't be. Expert 2's proof is fallacious. In the second case, the shortest path sometimes goes through two connecting vertices. An example in H3, pointed out by Lu [31], is shown in Figure 11. The identical m i s t a k e - - assuming that "the largest disc moves at
Figure 11. Lu's counterexample.
most once" when moving between any two positions by the most efficient r o u t e - - occurs many times in the literature on the Tower of Hanoi. See, for example, Refs. 30 and 33. Psychologists have used the tower of Hanoi as an experimental testbed for-human decision-making, for example, Ref. 34; and on occasion the same mistake has crept i n - - for example, in Ref. 35. Unfortunately, even when the nature of the fallacy is grasped, it seems hard to incorporate this third case into the analysis in the same manner as Figures 9 and 10, and the story becomes far more complicated. Hinz [36, 37], and independently Chan [38], give a formula for the average number of moves between positions in the Tower of Hanoi. In fact, the total number of moves (using shortest paths) between all possible pairs of positions is 46618 n 1 n 3 n 88---5 - 59 - 73
+4~-
100318v / ~ ) [ ~ ( 5 _ V , ~ ) ]
n
Thus, the average distance between two positions is asymptotic to (466/885)2 n. Hinz hadn't realised there was any connection with the Sierpifiski gasket; but having seen Ref. 15, he realised that the limit as n --* ~ of his result for n-disc Hanoi proves that the average distance between two points in a unit ,Sierpifiski gasket is 466/885 precisely (just normalize to make the diameter of the graph I, by dividing by 2 n - 1). This is some 2% smaller than the value suggested by Expert 2. Who says recreational mathematics has no serious payoff? At the moment, this approach via the tower of Hanoi is the only known method for finding the answer. For the statistically minded, Hinz also proved that the variance of the distance between two random points is precisely 904808318/14448151575. E n c o u n t e r 4: Barnsley's E n c o u n t e r w i t h S i e r p i f i s k i ' s G a s k e t ( M i c h a e l Barnsley, 1988)
A fractal set generally contains infinitely many points whose organization is so complicated that it is not possible to describe the set by specifying directly where each point lies. Instead the set may be defined by "the relations between the pieces". It is rather like describing the solar system by quoting the law of gravitation and stating the initial conditions. Everything follows from that. It appears always better to describe in terms of relationships. Michael Barnsley Fractals Everywhere In his celebrated textbook [39], Michael Barnsley introduces the chaos game. Mark three points in the plane, say at the vertices of an equilateral triangle. Obtain a threesided coin for which heads, tails, and edge have the same
Figure 12. The chaos game after 1000, 3000, and 6000 iterations.
probability, namely 89 Label the three vertices of the triangle correspondingly. N o w play the following game. Start with a randomly chosen point x0 in the plane. Toss the coin and move the point halfway towards the corTHE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
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responding vertex, getting a new point x~. Repeat this procedure, always generating Xn+l from Xn by tossing the coin and moving xn halfway towards the appropriate vertex. What do y o u see? You might expect the result to be some uniform cloud of points in the plane. But having read the title of this article, you might suspect that this is not so, and a shrewd guess would be ... well, Figure 12 shows that you're right. You get a Sierpifiski gasket, which becomes more a n d more sharply defined the more iterates y o u use. (The first 50 iterates are omitted, for these constitute dynamical "transients" that spoil the perfection of the figure.) This seems a very o d d shape to generate by a rand o m procedure, although we shall see shortly that it is entirely natural. Barnsley defines a generalization: an iterated function system or IFS. This is a finite set of affine maps from the plane to itself. Affine maps are specified by six parameters:
F(x, y) = (ax + by + e, cx + dy + f). The contractivityfactor of such a map is defined to be
an IFS, where Fn has contractivity factor sn, and suppose that Sn < 1 for all n. Define a set A to be invariant under ~v if n
a = U F~(A). i=1
For example, suppose maps F~ : • --~ R, (i = 1, 2) are defined by X
F~(z) = 5 '
F2(x) =
x+2 3
Then the standard middle-third Cantor set is invariant under Y'. T H E O R E M 3: Under the above conditions (in particular with all si < 1), there exists a unique nonempty invariant set for .F.
Proof:. Let 7-/be the set of all subsets of ~2 with the Hausdorff metric. Then ~- defines a contraction mapping on with contractivity factor s = max(sn). This has a unique fixed point. See Ref. 11 or 39 for details.
s = lad - bc[.
If s < 1, then F shrinks areas by a factor s. (If s > 1, it expands them by a factor s.) Suppose that .F = {F~} is
Figure 13. The black spleenwort fern.
62 THEMATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
In view of the proof, we denote the invariant set by Fix(Y). Typically, Fix(~-) is a fractal; and by definition, it is self-affine, that is, the union of affine copies of itself.
With certain technical hypotheses (the images of Fix(S-) u n d e r the fi should not overlap "too much"), the fractal dimension d of FixOv) is the unique value such that s~ + . - . + s~ = 1. The Sierpifiski gasket is obviously equal to Fix(Y) w h e n ~ = {F1, F2, F3} a n d
El(X, y ) =
( 2 2)
F2(x,y) = ( x q - 1 ~ ) 2 ~
F3(x, Y) =
(21
q- -~, 2 -]-
.
These three transformations correspond precisely to " m o v e halfway towards the vertices of an equilateral triangle." It is now clear that the point set defined by playing the chaos game is almost surely a very close approximation to the invariant set for the corresponding IFS, and that's w h y we see the Sierpinski gasket. Barnsley [39] and Falconer [11] contain the proof, plus extensive generalizations. This observation has a curious and potentially important consequence. Suppose y o u want to send a colleague a picture of the Sierpifiski gasket. You could d r a w one and run it through a fax machine. This will scan the page in raster fashion and send several hundred thousand numbers along the telephone lines, from which another ,fax machine can reconstruct the picture. On the other hand, if the recipient has a computer that can play the chaos game, all you need do is send the numbers that define the iterated function s y s t e m - - six per affine map, 18 altogether. This represents a considerable saving in data to be transmitted. As it happens, a great m a n y natural objects have fractal structure, and so can be given a "compressed" description as invariant sets of iterated function systems. So can m a n y nonfractals, such as a solid square (play the chaos game with four points and a four-sided coin). The traditional example is the black spleenwort fern (Figure 13). Although you might not often want to transmit a black spleenwort fern, most pictures are made up out of pieces that have the same kind of fractal structure, and an extension of the notion of an IFS can be applied to them: see Refs. 40--43 and 48. Initially the method was greeted with some skepticism, but it is a perfectly practical one: see Refs. 44 and 48. Commercial software to implement the process is a v a i l a b l e - - a t commercial prices. The whole story suggests a new view of c o m p l e x i t y - or at least, encourages a view more akin to algorithmic information theory [45] - - namely, it is the complexity of the process that produces an object that is important, not the apparent complexity of the object itself. Prescription, not description, is th,e key. It is a point of view with substantial implications for evolutionary and developmental biology; see Ref. 46.
Encounters, Encounters,... The variety of situations in which we encounter the Sierpifiski gasket is considerable. Indeed, there are m a n y more such encounters scattered throughout the mathematical literature: One I was told of recently is the graph of positions for hexaflexagons. These ingenious mathematical toys are described in Ref. 47. W h y do we meet the gasket in so m a n y different places? The underlying theme in all four encounters is recursion: The Sierpifiski gasket is the incarnation of recursive geometry. Indeed, it is probably the simplest genuinely two-dimensional recursive geometric object, just as the Cantor set is the simplest one-dimensional one. (I mean that the gasket lives in the p l a n e - - I ' m not talking about its fractal dimension.) Even given this rationalisation, it is still rather odd that it appears in so m a n y guises.
References 1. Benoit Mandelbrot, The Fractal Geometry of Nature, San Francisco: Freeman (1977). 2. W. Sierpifiski, Oeuvres Choisies (2 vols.), Warsaw: Pafiswowe Wydwnicto Naukowe, (1975). 3. Krzysztof Ciesielski, Lost legends of Lvov I: The Scottish Caf6, Mathematical Intelligencer 9(4) (1987), 36-37. 4. Krzysztof Ciesielski, Lost legends of Lvov II: Banach's grave, Mathematical Intelligencer 10(1) (1988), 50-51. 5. W. Sierpifiski, Sur une courbe dont tout point est un point de ramification, Compt. Rendus Acad. Sci. Paris 160 (1915), 302-305. 6. W. Sierpifiski, On a curve every point of which is a point of ramification, Prace Mat. Fiz. 27 (1916), 77-86 [Polish]. 7. W. Sierpifiski, Sur une nouvelle courbe continue qui remplit toute une aire plane, Bull Int. Acad. Sci. Cracovie A (1912), 462-478. 8. W. Sierpifiski, On a Cantorian curve which contains a bijective and continuous image of any given curve, Mat. Sb. 30 (1916), 267-287 [Russian]. 9. W. Sierpifiski, Un exeknple 616mentaire d'une fonction croissante qui a presque partout une deriv6e nulle, Giornale Mat. Battaglini (3) 6 (1916), 314-334. 10. W. Sierpifiski, On a reversible function whose image is dense in the plane, Wektor 3 (1914), 289-291 [Polish]. 11. Kenneth Falconer, Fractal Geometry, New York: Wiley (1990). 12. L.E. Dickson, History of the Theory of Numbers, Vol. 1, New York: Chelsea (1952). 13. P. Hilton and J. Pedersen, Extending the binomial coefficients to preserve symmetry and pattern, Computers Math. AppL 17 (1989), 89-102; reprinted in Symmetry 2 - - Unifying Human Understanding (I. Hargittai ed.), Oxford: Pergamon Press (1989). 14. Marta Sved, Divisibility--with visibility, Mathematical Intelligencer 10(2) (1988), 56--64. 15. Ian Stewart, Le lion, le lama et la laitue, Pour la Science 142 (1989), 102-107. 16. Ian Stewart, Game, Set, and Math, Oxford: Basil Blackwell, (1989) [reprint: Harmondsworth: Penguin Books (1991)]. 17. G. S. Kazandzidis, Congruences on the binomial coefficients, Bull Soc. Math. Grace (NS) 9 (1968), 1-12. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
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18. David Singmaster, Notes on binomial coefficients I I I - Any integer divides almost all binomial coefficients, J. London Math. Soc. (2) 8 (1974), 555-560. 19. J. W. L. Glaisher, On the residue with respect to pn+l of a binomial-theorem coefficient divisible by pn, Quart. J. Pure Appl. Math. 30 (1899), 349-360. 20. E. E. Kummer, Uber die Erg/inzungss/itze zu den allgemeinen Reciprocit/isgesetzen, J. Reine Angew. Math. 44 (1852), 93-146. 21. David Singmaster, Notes on binomial coefficients I - - A generalization of Lucas' congruence, J. London Math. Soc. (2) 8 (1974), 545-548. 22. A. W. E Edwards, Patterns and primes in Bernoulli's triangle, Math. Spectrum 23 (1991), 105-109. 23. Siegfried R6sch, Expedition in Unerforschtes Zahlenland, Neues Universarum 79 (1962), 93-98. 24. Siegfried R6sch, Neues vom Pascal-Dreieck, Bild der Wiss. (Sept. 1965), 758-762. 25. David Singmaster, Notes on binomial coefficients I I - - The least n such that pe divides an r-nomial coefficient of rank n, J. London Math. Soc. (2) 8 (1974), 549-554. 26. David Singmaster, Divisibility of binomial and multinomial coefficients by primes and prime powers, A Collection 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.
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of Manuscripts Related to the Fibonacci Sequence, 18th Anniversary Volume of the FibonacciAssociation (1980), 98-113. N. Claus [= Edouard Lucas] La tour d'HanoL jeu de calcul, Sci. Nature I (1884), 127-128. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, New York: Addison-Wesley (1989). R.S. Scorer, P. M. Grundy, and C. A. B. Smith, Some binary games, Math. Gaz. 28 (1944), 96-103. M. C. Er, A general algorithm for finding a shortest path between two n-configurations, Inform. Sci. 42 (1987), 137141. Lu Xuemiao, Towers of Hanoi graphs, Int. J. Comput. Math. 19 (1986), 23-38. D. Wood, The towers of Brahma and Hanoi revisited, J. Recreational Math. 14 (1981-82), 17-24. D. Wood, Adjudicating a towers of Hanoi contest, J. Recreational Math. 14 (1981-82), 199-207. E Klix, J. Neumann, A. Seeber, and H. Sydow, Die algorithmische Beschreibung des L6sungsprinzips einer Denkanforderung, Z. Psychol. 168 (1963), 123-141. H. Sydow, Zur metrischen Erfassung von subjektiven Problemzust/inden und zu deren Ver/inderung im Denkenprozet~ I, Z. Psychol. 177 (1970), 145-198. Andreas M. Hinz, The tower of Hanoi, L'Enseignement Math. 35 (1989), 289-321. Andreas M. Hinz, Shortest path between regular states of the tower of Hanoi, Inform. Sci., to appear. Chan Hat-Tung, A statistical analysis of the towers of Hanoi problem, Int. J. Comput. Math. 28 (1989), 57-65. Michael Barnsley, Fractals Everywhere, Boston: Academic Press (1993). Michael Barnsley, A Better way to compress images, BYTE, January 1988. Michael Barnsley and A. E. Jacquin, Application of recurrent iterated function systems to images, SPIE 1001 (1988), 122-131. A. E. Jacquin, A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, Ph.D. Thesis, Georgia Institute of Technology (1989). A. E. Jacquin, A novel fractal block-coding technique for digital images, ICASSP '90 (1990). Jon Waite and Mark Beaumont, An introduction to block based fractal image coding, preprint, British Telecom Research Station, Ipswich (1991). THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
45. Gregory J. Chaitin, Algorithmic Information Theory, Cambridge: Cambridge University Press, (1987). 46. Jack Cohen and Ian Stewart, The information in your hand, Mathematical Intelligencer 13(3) (1991), 12-15. 47. Martin Gardner, Mathematical Puzzles and Diversions from Scientific American, London: Bell (1961). 48. Michael E Barnsley and Lyman P. Hurd, FractalImage Compression, Wellesley MA: A. K. Peters (1993).
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom
Williams, continued from p. 34. W h e t h e r Daniel G u m b p r o v e d the theorem for himself or w a s s i m p l y so impressed b y the simplicity of the proof that he d e c i d e d to record it in stone is not clear. It is still easily recognisable despite 250 y e a r s of weathering. This proof is, of course, fairly well k n o w n . For example, it is m e n t i o n e d in Ref. 3 as "another proof." The cave is not v e r y easy to find. Figure 2 s h o w s it in relation to the Cheesewring (and the author) with the d i a g r a m easily recognisable on the roof. To those visiting this p a r t of Cornwall, and possibly considering the astronomical significance of the Hurlers, the location of Daniel G u m b ' s cave a n d proof is left as an exercise.
References 1. S. Baring-Gould, Cornish Charactersand Strange Events, Bodley Head (1909). 2. W.H. Paynter, Daniel Gumb, The Cornish cave-man mathematician, Old Cornwall:Journal of the Federation of Old Cornwall Societies, II (4) (1932). 3. C. Godfrey and A. W. Siddons, Elementary Geometry, 4th ed., Cambridge: Cambridge University Press (1962).
Faculty of Mathematical Studies University of Southampton Southampton S0171BJ, England
Figure 2
Jeremy J. Gray* Poincar4, Einstein, and the Theory of Special Relativity
acterised above all by the rule, that no velocity can exceed the velocity of light" (italics Whittaker's).
To see how from these insights "the whole science of physics [was] reformulated in accordance with Poincar6's Principle of Relativity," Whittaker then reviews the work of Lorentz. He argues that Lorentz was led to a theory, exact to all orders in v,/c, in which "he replaced the condition of transforming (x2 +.y2 + Z2) into itself, by the condition of transforming (x 2+ y2+ z 2_ c2t2) into itself." That these transformations form a group was one of the achievements of Poincar6 in 1905. Poincar6 also extended Lorentz's analysis to show that Maxwell's equations are invariant with respect to the Lorentz transformations. Then, comparing the work of Poincar6 and Lorentz, Whittaker concludes that "it was Poincar4 who proposed the general physical principle." A few pages later, Whittaker summarises an earlier paper of Poincar6 this way: "In 1900 Poincar6 ... suggested that electromagnetic energy might possess mass density equal to (1/c 2) times the energy density; that is to say, E = mc 2, where E is energy and m is mass. In 1905 A. Einstein ... suggested the general conclusion, in agreement with Poincar6, that the mass of a body is a measure of its energy content." By now, it should come as no surprise to read that in 1905 "Einstein published a paper which set forth the special relativity theory of Poincar6 and Lorentz with some amplifications, and which attracted much attention." Or rather, it comes as a great shock, for despite the longevity of Whittaker's views, they differ widely from the generally accepted opinion, which attaches Einstein's name alone to the theory of special relativity. Can it really be that the saintly white-haired image has eclipsed a much less remarkable young Patent Officer in Bern? At Nancy, Jules Leveugle gave his account, published in the issue of La Jaune et la Rouge [2] dedicated to the bicentenary of the I~cole Polytechnique) He goes over much of the same material as Whittaker, whose work he acknowledges, and amplifies it at some points. For example, he compares Poincar6's discussion of how clocks must be used, given in the St. Louis address, with Einstein's. He argues also that Poincar6 rejected the ether.
* Column Editor's address: Faculty of Mathematics, The O p e n University, Milton Keynes, MK7 6AA, England.
1 Whence the reference to 1873; Leveugle himself belongs to the class of 1943.
One of the most persistent stories that circulates in the communities of historians of mathematics and historians of science is that Poincar6 came to the theory of special relativity before Einstein. The prime written source for this claim is E.T. Whittaker's A History of the Theories of Aether and Electricity [1], which was first published in 1953. The latest sighting of which I am aware was at the Colloque Henri Poincar6 held in May 1994 in Nancy. (The Colloque was held as part of the moves to establish the new archive of Poincar6 material there and to organise the publication of Poincar6's still-unpublished papers and correspondence.)
Whittaker's Claim ,Briefly, Whittaker's claim is that the theory of special relativity is the creation of Poincar6 and the Dutch physicist H.A. Lorentz. He therefore refers systematically to the Poincar6-Lorentz theory of relativity: the relevant chapter, Chapter II, is, indeed, called "The Relativity Theory of Poincar6 and Lorentz." Whittaker begins by reviewing the failure of experiments to detect the velocity of the Earth relative to the supposedly all-pervading ether [to first or second order in the ratio of the velocity of the Earth (relative to the ether) to the velocity of light]. This led Poincar6 in lectures at the Sorbonne to the idea that "optical phenomena depend only on the relative motions of the b o d i e s . . , concerned." Whittaker interprets this as meaning that "Poincar6 believed in 1899 that absolute motion is indetectible in principle" (italics Whittaker's). In 1900, Poincar6 went on to claim that, in fact, the motion of the Earth relative to the ether should in principle be indetectible. Then, in his St. Louis address of 1904, Poincar6 proposed the principle of relativity: "The laws of physical phenomena must be the same for a 'fixed' observer as for an observer who has a uniform motion of translation relative to him." From this he deduced that "There must arise an entirely new kind of dynamics, which will be char-
THE MATHEMATICALINTELLIGENCERVOL.17, NO, 1 (~)1995Springer-VerlagNew York 6 5
He cites variously from Poincar6's La science et l'hypoth~se [3], which represents Poincar6's views between 1899 and 1902. From Chapter 6, "La m6canique classique" (pp. 111-112): "There is no absolute space . . . . no absolute time .... no direct intuition of the simultaneity of two events produced in different places." From Chapter 7, "Le mouvement relatif et le mouvement absolu" (p. 133): "This does not contradict the fact that absolute space, that is to say the frame to which we must refer the earth to know if it really turns, does not exist." From Chapter 12, "Uoptique et 1' electricit e" (p. 215): "It scarcely matters if the ether exists, this is a matter for the metaphysicians; what is essential for us is that everything happens as if it exists and that this hypothesis is useful for explaining the phenomena." Leveugle's essay has an almost legalistic character: The theory of special relativity consists of these ideas; my client, M r Poincar6, had all of them before anyone else; therefore, m'lud, the theory is rightly his.
The Case for Einstein The case for Einstein was put at Nancy by Professor A.I. Miller, who has written about it at length over the years. Insofar as it concerns Whittaker, he published his rebuttal [4] in 1987, noting his disagreements with Whittaker in his preface to a reprint of Whittaker's book. In contrast to "the meticulous literature citations elsewhere in these two volumes," he finds that Whittaker's Chapter II "is fraught with substantial historical errors." First, he notes that Poincar6 accepted Lorentz's theory of 1895 as the best available, because it dealt adequately with all experiments, but he criticised it for violating Newton's principle of action and reaction. It was to deal with this problem that Poincar6 wrote his paper of 1900 cited above. There, Poincar6 proposed the formal analogy that "electromagnetic energy behaves ... as a fluid endowed with mass." From this, the recoil velocity of an emitter of a particle of light can be written down. But "at neither this point in his 1900 paper, nor at any other place, did Poincar6 write down E = mc 2 for electromagnetic energy, where m is a mass to be associated with a real electromagnetic field energy." Moreover, according to Miller, only Einstein, but not Poincar6, discussed "the equivalence of mass and energy for any sort of energy, including the conversion of mass into energy in radioactive decay processes" (italics Miller's). In 1906, Einstein agreed that his results on mass-energy equivalence were "in principle" contained in Poincar6's paper of 1900 but asserted that he had come to his conclusion independently. Indeed, Miller argues, his conclusions are, in fact, more general. As for the invariance of (X 2 q- y2 _}_ Z2 __ c2t 2) rather than of (x2 + y2 _j_ Z2) as being a key perception of Lorentz, Miller argues Lorentz did no such thing. Rather, the "unwieldy statement in 1904 entailed three different coordinate systems, which led Lorentz to commit se66 THEMATHEMATICALINTELLIGENCERVOL.17, NO. 1, 1995
rious mathematical errors," errors that led Poincar6 to set Lorentz's theory on a firm foundation. That done, Poincar6 could not only correct Lorentz's mathematical errors, he could also resolve a serious problem with the physics. Lorentz's electron was unstable, owing to its Coulomb self-field. Poincar6 resolved this by introducing what are nowadays called Poincar6 stresses, nonelectrical forces which enable the conservation of energy and momentum laws to be applied consistently to determine the electromagnetic mass of the electron. Later, Max von Laue, building on contributions from Minkowski, showed that Poincar6 stresses enable one to prove that the energy and momentum of an electron transform correctly under Lorentz transformations (Feynman, Lectures in Physics, 1964, II-28-4). This achievement is not discussed by Whittaker. What of the St. Louis address? To the claim that Poincar6 postulated a principle of relativity at the heart of a physics in which no velocity exceeds that of light, Miller replies that, for Poincar6, the principle of relativity "holds exactly for Newton's mechanics and appears in the portion of the lecture pertaining to 'Newtonian mechanics.'" When the dynamics of the electron is at stake, matters are less clean Poincar6 wrote (I prefer Miller's translation to Whittaker's), "From all these results would arise an entirely new mechanics which would above all be characterised by the rule that no velocity could exceed the velocity of light" [5 (VS 197)]. This is far from claiming to have such a thing. In the famous Palermo paper of 1905, surely the definitive statement of Poincar6's position, Poincar6 hedged his bets precisely because Walter Kaufmann had recently announced experimental results that cast doubt on Lorentz's theory, the very theory Poincar6 was trying to defend. It is clear from his writings how Miller would reply to Leveugle's claim that Poincar6 abandoned the ether, for he quotes a remark of Poincar6 from Ref. 6: "Beyond the electrons and the ether there is nothing." This prompts us to look more carefully at the texts themselves. As the quote from 1908 shows, the first thing one notices is their inconsistencies. To some extent, this is a feature of Poincar6's philosophy of conventionalism, which prevents him from coming out against any kind of useful hypothesis. For him, two systems of ideas were paramount: the results of experiments (when reliable) and a small number of mathematical laws, such as conservation of energy. No genuine significance attached to one set of variables rather than another, and therefore to one set of physical quantities rather than another, provided that the mathematical theory led to the known experimental laws (usually expressed as differential equations). This philosophy admits a good deal of equivocation. More worrisome for the Whittaker thesis, when discussing problems with the principle of relativity, Poincar6 speaks of signals that do not travel at the speed of light and says, "And are such signals inconceivable, if one admits with Laplace that universal gravitation is
transmitted a million times faster than light?" (VS 189). This in the same speech where, it is said, he most effectively prefigured Einstein. These wayward texts also point in the opposite direction, however. Miller cannot confine Poincar6's principle of relativity to Newtonian mechanics. Poincar6's conclusion, after reviewing the principle of relativity in the light of problems in electrodynamics is in fact that it survives: "the principle of relativity has been valiantly defended in recent times, but even the energy of the defence shows how serious was the attack" (VS 190). He concluded that, "Perhaps therefore w e must construct an entirely new mechanics which we can only glimpse, where, inertia increasing with velocity, the velocity of light becomes an insurpassable limit." (VS 210) H o w can such a remarkable discrepancy between experts come about? A clue comes from the most trenchant of Miller's points, which is directed at Whittaker's claim that it was Poincar6, and not Lorentz, who was the leading physicist of the two. It is not that it is wrong, as even a cursory look at the papers shows, says Miller, but that it is misconceived. Poincar6 and Lorentz were trying to understand the electron by creating a unified theory of electromagnetism and gravitation. It was Einstein's original idea to re-create our theories of space and time. Only by not seeing that Einstein changed the terms of the debate can the whole enterprise of putting together a series of partial insights and hypotheses be presented as the first occurrence of a remarkable theory. One may speculate on w h y Whittaker got it just so ex"travagantly wrong. The answer is not likely to lie in a failure to understand the intellectual issues; Whittaker was one of a number of British mathematicians well versed in mathematical physics. Nor was he a stranger to scholarship; his attention to the literature would put many a m o d e m mathematician to shame. It would be interesting to look at other British writers on this subject and see where they stood, but in any case, as Miller points out in a footnote, it is disturbing to see Whittaker repeat in 1953 some of the main features of the Nazi line on Einstein. Probably the explanation lies at some cultural level, but it may simply be that, writing in his seventies, Whittaker was attracted to a position just because it was startling and, if correct, would debunk a modern myth. Contemporary reviewers may indicate something of Whittaker's startling position. All were impressed with Whittaker's erudition. P.W. Bridgman, writing in ISIS, spoke of one's "stupefaction at the industry and versatility of the author"; Freeman Dyson in Scientific American called the book "the most scholarly and authoritative history of its period that we shall ever get." But Dyson, while noting that he could easily have ascribed many of the quotes from Poincar6 to Einstein, professed himself unable to pass judgement upon the historical accuracy of the book. Bridgman alluded to much forgotten history, in particular the "little-known pre-history of the massenergy relation." It was left to Max Born, publicly and
privately, to defend Einstein. The disciplinary side of the matter surely also plays a role. Few have brought to bear the skills in both physics and mathematics that Whittaker could command. Einstein has belonged to physicists and the historians of physics, but even Pais, the best of his biographers, sees little of Poincar6. Mathematicians, on the other hand, find it hard to master all the details of physical theories rendered obsolete by special relativity, even though it is in the physics that Poincar4's originality lies and where, by the standards of Einstein, he fell short. They see the Lorentz transformations but not the meanings they could carry. The last words may, however, rest with Poincar6 a n d Einstein themselves. One may ask how these two saw each other's achievements, for both were capable of generosity and honesty. They saw very little. Although he wrote on the theory of special relativity, Poincar6 never mentioned Einstein by name, nor did he ever associate Einstein's name with Lorentz's theory of the electron. And Einstein said of tris one recorded rn'eeting with Poincar6, at the Solvay conference in 1911, that he was antagonistic to the theory of relativity and showed little understanding of the situation [7]. In the face of such disagreement, it is surely not likely that Einstein's theory of special relativity was really the Poincar4--Lorentz theory all along. A Grumpy Endnote The question of checking the quotations is, sadly, worth a comment. It is not just that writers in English tend to rely on existing English translations, often the handy Dover editions, with few references to what Poincar~ originally wrote. Rather worryingly, Whittaker gave very few precise page references. French writers, on the other hand, refer to the French editions of Poincar6's books of essays, which exist in several editions, making page references (when given) often unhelpful. Moreover, the essays themselves were often published in journals first and may well have been modified before being included in the book version. In addition, they were composed at different times, so the dat~ of publication of the book may give a misleading impression of the state of Poincar6's fast-changing ideas. A one-volume reedition of all the essays (including the few not previously anthologised) with a short critical commentary would be useful. J. Vuillemin's reedition of La science et l'hypoth~se in 1968 [3] is indicative of what can be done. For the record, Poincar6's St. Louis address, "UEtat actuel et l'avenir de la physique math6matique," was first published in Bull. Sci. Math. (2) 28 (1904), 302-324. It was then reprinted in La valeur de la science where it was broken into three essays or chapters and given occasional paragraph headers. References are given to it here in the form (Bull 306 - VS 176). S~ far as I can tell, the essay is otherwise identical. Continued
o n p . 75
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67
Jet Wimp*
Functions and Graphs by I.M. Gelfand, E.G. Glagoleva, and E.E. Shnol Boston: Birkhafiser, 1990. ix + 105 pp. US $16.50, ISBN 0-8176-3532-7
The Method of Coordinates by I.M. Gelfand, E.G. Glagoleva, and A.A. Kirillov Boston: Birkhaiiser, 1990. ix + 73 pp. US $16.50, ISBN 0-8176-3533-5
Algebra by I.M. Gelfand and A. Shen Boston: Birkhafiser, 1993. 153 pp. Softcover, US $18.50, ISBN 0-8176-3677-3 Hardcover, US $24.50, ISBN 0-8176-3737-0
Reviewed by H. Wu "The most important thing a student can get from the study of mathematics is the attainment of a higher intellectual level." So wrote Gelfand in the common preface to the first two books under review (FG and MC for short). It may seem truistic in academic context, but anyone who has spent 2 years surveying the contemporary scene in mathematics education and has had more than his or her share of hyperboles such as "mathematical empowerment of the students" or "political development" in the current reform just might find Gelfand's simple statement refreshing, nay, moving. A mathematician preoccupied with his own research and his daily duties may not be aware that there is a reform underway in the mathematics education of K-12 (i.e., kindergarten through the 12th grade) in the United States. In the eighties, many educators rightly felt that the traditional mathematics instruction in the schools had degenerated into a ritual, one that no longer had relevance to either mathematics or education. This was eas-
* C o l u m n Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA.
68
ily verifiable through the poor performance of the high school graduates in the high-tech work force. As a result, the business c o m m u n i t y - - a m o n g many sectors of society-- started to agitate for improvement in the quality of mathematics education. There was also the matter of massive dropouts and abysmal test scores in mathematics, and the statistics were, of course, grist for the political mill. For example, when George Bush was campaigning for the presidency in 1988, he saw fit to adopt as a campaign slogan that he, if elected, would be our "education president" and would make the science and mathematics education of this country the first in the world by the year 2000. Subsequently, President Clinton took over this theme and has recently signed the Goals 2000 legislationJ With all these forces at work, attempts at reform became inevitable. In fact, the reform effort has also spread to the teaching of calculus in college. The National Science Foundation has spent millions to sponsor the development of mathematics curricula for both schools and colleges in line with the proposed reform. The prospect is for an exciting era in education. The publication of the NCTM Standards [1] in 1989 marked the beginning. This document has since become the rallying point in any discussion of the reform. Although it may be too early to assess the achievements of this effort, there are good reasons to assess the implications of some already recognizable trends. For the purpose of this review, I will limit myself to a brief report, based on the publications available to me, on how the reform movement has affected the content of the mathematics curriculum in 9-12 and calculus. It is to be noted that the reform addresses not just content but also the method of teaching (e.g., the stress on group learning in the classroom and the integration of calculators and computers into the instruction) and the method of assessment (e.g., in the current educational jargon, the emphasis on "process" over "product"). These are even more controversial. The self-imposed restriction to a discussion of the 1 In the case of President Clinton, however, one should think twice before ascribing the motive of political expendiency to his action, because his track record as governor of Arkansas and his interviews on the subject of education show him to be uncommonly well informed and dedicated to this cause.
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1 ~)1995 Springer-Verlag New York
content is nothing more than a sensible decision to get this review written in time. In general terms, the reform movement has forced m a n y teachers to rethink their day-to-day teaching, to loosen up their previously rigid classroom atmosphere, to jettison some musty topics and ugly technical formulas, and to pay more attention to the student's needs. As with almost anything new, it can be to some a breath of fresh air. The resulting enthusiasm can be seen in the way the NCTM Standards is now embraced by an overwhelming majority of the educators, as well as a sizable portion of the mathematicians connected with education. It may seem churlish, therefore, to say at this point that all is not well with the content of the proposed reform and that a more measured response to these innovations is in order. Nevertheless, let me attempt such a response. I shall concentrate on only three areas. One major emphasis of the current reform is on the "process" in mathematics rather than the "product." Roughly, this means more stress on the general, qualitative reasoning of mathematics at the expense of technical skills and neat formulas.2 This is clearly a reaction to the mindless and excessive routine computations that characterize many of the elementary texts. The very formalsounding textbooks still in use in most classrooms are now being replaced by books filled with heuristic arguments, conjectures, and examples. There is no question that this is welcome, but one must ask whether the "process" is backed up by solid mathematics. Qualitative reasoning is important but so are precise formulas and Iong computations. Many mathematicians have pointed out that, in response to having "technique" replaced by "technique-without-understanding," the reform movement has now gone to the other extreme of deemphasizing basic techniques and thereby gutting mathematics. Thus, we find in this climate a 9-12 curriculum which allocates the quadratic formula only to the college-bound students in the 12th grade [2], a precalculus text that does not do the binomial theorem or the geometric series [3], and a beginning calculus text that does not treat the convergence of infinite series or l'Hospital's rule [4]. A second major emphasis of the reform is on the general issue of "relevance," that is to say, the role of mathematics for solving everyday problems. This has to be understood literally. "Applications" in the past used to be synonymous with "applications to the physical sciences," for example, the deduction of Kepler's three laws from Newton's inverse square law. Now, "applications" means largely statistical phenomena directly related to social issues or tangible problems of our everyday lives, for example, how to set the "bast" speed limit on a given
2One may be forgivenif one hears in this the echo of Tom Lehrer's classicput-downof TheNew Math: "Youtakesevenfromthirteenand that leaves five,well, the answer is actuallysix, but it is the idea that counts."
stretch of freeway, or compute the height of a rider on a Ferris wheel as a function of time [2, 3]. The literature of the curriculum development project ARISE (which has been funded by the National Science Foundation to develop a complete mathematics curriculum for 912) states, for example, that "In ARISE, the mathematics truly arises out of applications. The units are not centered around mathematical topics but rather application areas and themes, with the mathematical topics occurring as strands throughout the unit" [5]. Whether or not this group is proposing to inculcate the idea in high school that "mathematics = industrial and applied mathematics" I leave for the reader to decide. An obsession with applications can lead to a mathematics curriculum without mathematical cohesion or structure and to a mathematical education that does violence to mathematics as a branch of knowledge that stands on its own. In addition, one can question the pedagogical value of the current deemphasis of applications to the physical sciences. Insofar as applications are supposed to demonstrate the-power of mathematics, there is no doubt that those arising from the physical sciences do so most convincingly. They are also the ones that carry the most potent mathematical ideas and have the further advantage that the implication of the mathematical outcome rarely involves any uncertainty of interpretation. A third emphasis of the reform movement is on minimizing the role of proofs in the regular curriculum. Because "proof" is, at present, a slightly obscene word in mathematics education, an even-handed approach to this issue must begin with the fact that proofs were never accorded their rightful place in the older (traditional) curriculum. Should the reader have any doubts, a casual perusal of almost any algebra text currently in use in the schools would dispel them. What happened in the past was that, when all else failed, one could always count on Euclidean geometry to give the students a modicum of precise logical thinking. Unfortunately, the mathematical training of the average teacher could not (and cannot) be trusted to giv~ adequate instruction in twocolumn proofs, and the resulting courses in geometry tended to be a travesty.3 Given this reality, the hostility towards proofs in the education circle should come as no surprise. One unfortunate example of this extremism in mathematics education is a popular geometry text, highly praised by many teachers, with essentially all the proofs omitted [6].4 Another one is a whole curriculum
3 The p r o b l e m of teacher qualification, or rather the lack thereof, is a central one in the current " m a t h crisis" in t h e U n i t e d States. It is e a s y to b l a m e the teachers for this problem until one realizes that it is the m a t h e m a t i c i a n s w h o train the future teachers a n d that it is the reward s y s t e m of o u r society w h i c h indirectly selects them. There is, thus, e n o u g h b l a m e to be shared by all concerned. Unfortunately, we cannot go into this p r o b l e m here, as it w o u l d take a full treatise to do it justice. 4 Discussion of proofs in this 694-page text b e g i n s on p. 563 (and is b a d l y done), b u t the s t u d e n t s almost n e v e r get to p. 563. THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1,1995 69
with absolutely no proofs [2],5 which is justified by the statement: " . . . s e c o n d a r y school is [not] the place for students to learn to write rigorous, formal mathematical proofs. That place is in u p p e r division courses in college" [7]. Yet another is a calculus text that does not prove a single theorem [4]. In all these examples, heuristic arguments are routinely given, and some of them are correct proofs. However, as the latter are not clearly separated from others that are logically incomplete or e v e n invalid, students never learn w h a t a proof is. The kind of confusion and abuse such a mathematical education leads to is easy to imagine; for the record, see Ref. 8 for examples. An independent observer m a y be surprised b y the inherent contradiction in the simultaneous emphasis on process in mathematics and the de facto b a n i s h m e n t of proofs from the curriculum. Perhaps the reform has m a n y concerns, and that of making mathematics accessible to all students overrides all others. Seeing that the art of formulating a correct mathematical a r g u m e n t is not one of universal appeal, some educators w e r e probably p e r s u a d e d to take the line of least resistance. Be that as it may, the current r e f o r m m o v e m e n t is definitely kinder to the students in the lower half than those in the top 20% (say), and the question of h o w to take p r o p e r care of students serious about learning mathematics is left unresolved for now. 6 By coincidence, the D e p a r t m e n t of Education issued a remarkable d o c u m e n t [10] only 6 months ago which discusses in d e p t h h o w the American schools have failed to educate the talented students. It is difficult not to see Ref. 10 as a reproach of this senseless drift in mathematics education, from allowing proofs to be m e m o r i z e d without u n d e r s t a n d i n g to essentially denying the students the o p p o r t u n i t y to learn a b o u t proofs altogether. One m a y ask h o w a well-intentioned d o c u m e n t advocating reform such as the NCTM Standards [1] could go so wrong. The truthful answer is that the N C T M Standards was written to be all things to all people. On almost every issue, it is o p e n to m a n y interpretations. In fact, one educator was m o v e d to remark that NCTM wants to avoid imposing a curriculum on the teachers so that they w o u l d take more initiatives in interpreting and implementing the goals suggested in the Standards. So what are these goals? In general terms, N C T M wants the students to learn to think, and it also wants them to learn "significant mathematics." It is easy to agree with such noble ideals until one realizes that, w i t h o u t precise instructions on the content, pedagogy, and assessment
5 See the comment below about heuristic arguments. 6 See Ref. 9 for a discussion of this issue. The question of why the reform movement would not openly advocate the establishment of a system with built-in choices,whereby the students choose among two or more kinds of courses on the same subject which are differentiated by the amount of technical emphasis in each, is one that has never been answered without social-engineeringjargon. 70
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
of such a curriculum, and especially without a corps of mathematically competent teachers to implement it, they are not achievable goals in the United States in 1994. It appears to at least one observer that the tone set by the NCTM Standards has everything to do with the developments detailed in the preceding paragraphs. In its large 256 pages, the Standards makes repeated references to the value of a mathematics education in the schools as a valuable tool to earn a living in the high-tech work force. It also overemphasizes the applications of mathematics to the social sciences and e v e r y d a y life, as previously discussed. By contrast, it only mentions in passing (on p. 5) the n e e d to learn "to value mathematics" in a cultural and historic context; that is almost the last time the words "culture" and "history" m a k e an appearance in the book. Given this glaring imbalance, a reader of the Standards is not likely to associate learning mathematics with "the attainment of a higher intellectual level." Such a hard-nosed pragmatic approach to mathematics is b o u n d to s p a w n anomalous activities. The deleterious effect of the current reform on the mathematical component of e l e m e n t a r y mathematics education s h o u l d be a matter of grave concern to all mathematicians. Yet most are not even aware of the reform, and among those few that are, a majority seem to be enthusiastic about the reform itself as well as about the NCTM Standards [1]. As a consequence, the voice of dissent, so vital to a n y intellectual enterprise, is, thus far, largely absent. A l t h o u g h a recent letter [11] and an article [12], both extremely critical of Ref. 1, did make it to the pages of Notices of the American Mathematical Society, their tendency to overstate their cases and the intemperance o f the language give the readers the false impression that only extremists are resisting the reform movement. The mathematical c o m m u n i t y w o u l d be very poorly served indeed if it failed to get the message that a real crisis in mathematical education is looming on the horizon.
The Books Under Review All three books are a throwback to the bygone era w h e n a student's mathematical achievement was judged by fairly objective standards and the notion of learning for its o w n sake was still greeted with some approbation. The Foreword to FG says, "This book as well as others in this series is intended to be compatible with computers. H o w e v e r , . . . (the) c o m p u t e r c a n n o t - - nor will it ever be able t o - - think and u n d e r s t a n d like you can." By and large, these books challenge students to understand mathematics on its o w n terms; no beautiful computer graphics (in particular, no pictures of fractals), no fancy display of c o m p u t e r power, and no jazzy real-world applications. With this understood, w h a t these books manage to accomplish is to give a surpassing demonstration of the art of mathematical exposition that falls outside the prescriptions of the NCTM Standards.
thinking process that leads up to the correct definition of a concept or to an argument that clinches the proof of a theorem. Most, if not all, beginning students are sorely in need of this material, as are m o s t teachers. We are, therefore, v e r y fortunate that an account of this caliber has finally m a d e it to the printed page. Let me give an example: the discussion of the unit cube in ~4 in Part II of MC. If m y o w n experience is at all typical, for most of us, our first encounter with n dimensions was tentative and not a little tinged with anxiety. We more or less equated n with 3, as w e were told to do on an intuitive level, and w e secretly h o p e d that this oversimplification w o u l d not lead us astray. N o w enters MC, which shows us that this transition from 3 to n in fact can be methodical, smooth, and (to use a m u c h abused word) fun. It leads the reader gently to four dimensions b y first carefully examining the unit cubes in dimensions 1, 2, and 3, and then using all the information so accumulated in the visible world to extrapolate to dimension 4, again carefully and painstakingly. It looks at the situation not only geometrically but also analytically and makes sure that the reader can correlate the information in these two separate domains. It counts the vertices, edges, and faces of the cubes first in each of the visible dimensions, so that w h e n it comes to dimension 4, the extension of the counting to the invisible cube b y analogy becomes almost effortless. I dare say that a n y o n e w h o has taken FG and MC this guided tour will never be intimidated b y dimension A sine qua non of a book about mathematics is that it n ever again. So the only question is, W h y hasn't any of ,be mathematically correct. Although this sounds trivial, us thought of writing something like this? This is the kind of basic mathematical thinking that the fact remains that m a n y elementary textbooks contain serious errors, s Given the stature of the authors in the all high school students and their teachers should be present case, one can take for granted that these books exposed to, i n d e p e n d e n t of the considerations of "realare mathematically correct. Beyond correctness, a book world application," "technology, .... relevance," or w h a t can impress by its good taste in the selection of materials not. A n y mathematics education reform should make evor the choice of a particular approach to a topic. Or it can ery effort to ensure that the students (and their teachers impress by the pellucid style of presentation of a subject as well) have access to this kind of writing. Incidentally, I used the w o r d "painstakingly" above. If that easily becomes abstruse in lesser hands. An elementary book can also impress b y the inclusion of u n e x p e c t e d this conveys the impression of dullness and p e n d a n t r y insights along a m u c h t r o d d e n path. These two books are in the exposition of MC, then I w o u l d like to assure the impressive for all the above reasons, but more is true. To reader that exactly the opposite is true. The exposition me, their most striking characteristic is the a m o u n t of is lively and charming throughout. For other felicitous space devoted to a clear and detailed exposition of the examples, take the experimental approach to the focusinner workings of mathematics: All through both vol- directrix p r o p e r t y of the parabola on p. 42 of FG, the umes, one finds a careful description of the step-by-step motivation for the definition of the tangent of a curve on p. 77 of FG, and the discussion of the wings of a butterfly on p. 52 of MC. Every b o o k has its flaws, and these volumes are no 7 One can get further information by writing to Harriet Schweitzer, exception. There are two that are obvious and pressAMCS, CMSCE,SERCBuilding,Room 239,Bush Campus, Piscataway, ing: They should have an index and they should clearly NJ 08855-1179,or e-mail harriet@ga~dalf.rutgers.edu. 8Let me give one example. The usual treatment of sine and cosine in specify the precise knowledge a s s u m e d of the reader. calculus texts assumes that one can compute the length of an arc in the (It seems that MC should precede FG and that Algebra unit circle so that the radian measure of an angle can be defined. But should precede both.) of course, arc length has not yet been defined up to that point, so that Although the easy-going and conversational tone of the usual "proof" of the "Theorem" that d sin x/dx = cos x purports the exposition reflects faithfully the smooth progression to prove something about an object not yet properly defined. The use of the word "proof" rather than "heuristic argument" in this context is of the mathematical ideas, there are major discontinuities in three places. Unfortunately, these come without then an error (although it is easy to fix). FG leads the reader t h r o u g h the first steps of graphing simple functions: linear functions, y = Ix], quadratic functions, fractional linear functions, and p o w e r functions. MC gives a leisurely tour of the coordination of the line, plane, 3-space, and 4-space. They follow the s o u n d pedagogical principle that if students u n d e r s t a n d the simple concrete cases thoroughly, the extrapolation to the general situation will not be difficult. The slimness of b o t h volumes gives the correct indication that each has a well-defined and quite limited objective. For this reason, they w o u l d most likely be used as s u p p l e m e n t a r y materials in the (American) classroom, and we shall discuss t h e m accordingly. On the other hand, Algebra is a more ambitious book and is considered b y some as a potential textbook. It traverses a m u c h more extensive mathematical terrain. Of necessity, the review of this book will have a different focus and m u s t be done separately from FG a n d MC. These three books are used as texts in the Gelfand International Mathematics School (a correspondence school7). Three other volumes have been promised in the same series (Calculus, Geometry, and Combinatorics). For the most part, the following will concentrate only on the relation of the three books u n d e r review to American high-school education.
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
71
warning. On p. 82 of FG, the reader is asked to believe what amounts to lim
x--*•
x-1 = 0. 2x + 1
X 2 q-
For the intended readers of this volume, some kind of discussion or a more careful argument should be in place to signal the transition to something of greater subtlety. If the decision was not to enter such a discussion, then some disclaimer to this effect would also serve the purpose. Instead, what one gets is "business as usual," and this might just baffle the unsuspecting reader. Next, p. 48 of MC gives a beautiful argument to count, asymptotically, the number of lattice points N in a circle of radius v ~. It reads, "Thus we get the approximate formula N ~ ~rn." The problem here is that the symbol "~-," is never defined and there is again no warning about the jump in mathematical sophistication at this juncture. Furthermore, the subsequent argument justifying this formula on p. 49 is on a higher level than the informal tone would seem to indicate. Either it should be greatly expanded, or some cautionary statement to the reader is in order. Finally, Figure 34 on p. 70 of MC gives a three-dimensional projection of the four-dimensional cube. The paragraph in the lower half of the page purports to explain h o w that is done, but its terseness is not in line with the very detailed discussion up to that point. MC has many passages in fine print. It also has some traffic signs (?) next to certain passages. A little explanation of these interesting conventions would be gratefully accepted by the readers, even at the risk of lowering the CQ (Charm Quotient). High school students (or teachers) reading through these two books would learn an enormous amount of good mathematics. More importantly, they would also get a glimpse of how mathematics is done. This is an example of how the stated goals of the NCTM Standards (learn how to think and learn significant mathematics) can be fulfilled without the use of group learning or technology and without any extraneous need to make mathematics relevant to daily life. In particular, the detailed presentation of the thinking process in these books allows the readers to discover with the authors something new at each step. There we have the "discovery method" in action without the usual associated educational paraphernalia. True, these books may not be for everyone, but then nothing ever is. If this shows anything at all, it is that there are diverse and equally valid approaches to learning.
Algebra The topics treated in this volume constitute a good part of what is commonly known as Algebra II in the American schools: review of the basic properties of the ring of integers, including the division algorithm; raising numbers to integer powers; expansion of (a + b)n and 72
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1, 1995
Pascal's triangle; polynomials and rational expressions; arithmetic and geometric progressions; geometric series; quadratic equations; roots and noninteger powers; inequalities and the inequality of arithmetic and geometric means; quadratic means and harmonic means. The qualities of incisiveness, insight, and impeccable taste that set FG and MC apart from other books of the same genre also infuse the present volume. Were Algebra to be used solely for supplementary reading, it could be wholeheartedly recommended to any high school student or any teacher. (But note the discussion of some flaws below.) In fact, given the long tradition of mistreating algebra as a disjointed collection of techniques in the schools, there should even be some urgency in making this book compulsory reading for anyone interested in learning mathematics. They would discover, perhaps for the first time, that algebra is a logically coherent discipline, and at the same time that its formalism is a product born of good sense and sensible conventions. Both may come as a surprise. If one tries to give substance to the preceding general recommendation by specific examples, one is confronted by an embarrassment of riches: Where to begin? One might begin by highlighting the down-to-earth character of the whole book. On p. 11 and again on pp. 1517, there are cogent explanations of the need for symbols; such a discussion is probably not to be found in any other book of this level. Consider next the perhaps perplexing notation a -'~ for a positive integer n. It may occur to a beginner to ask w h y anyone would bother to write this instead of 1/a n. Basically, the answer is that once this notational convention is adopted, it would start to "think for us." Look at something different now: the division algorithm for polynomials. Instead of giving the formula and the relevant definitions right away, the book first performs three concrete divisions with lowdegree polynomials and describes the algorithm in complete sentences, without symbols. A final example: Before presenting the Pascal triangle and its basic property ((n~-l) _4_(in~l)= (i+l) ) , the book examines in detail th e expansion of (a + b)'~ for n = 2, 3, 4, 5, verifies the property in each case, and points out the pattern underlying this property. Then when the property is finally stated in full generality (p. 43), it becomes no more than an afterthought. (It is done without using the notation of the binomial coefficients.) The next class of examples have to do with a serious mathematical issue: the question of existence and uniqueness. Although these concepts undoubtedly lie at the core of mathematics, they would seem to be without any "relevance" whatsoever by the current standard of the reform movement. The book touches on one or the other more than once: for example, the existence and uniqueness of the (positive) square root of a positive number a (p. 98), the uniqueness of the quotient and remainder in the division algorithm for polynomi-
als (p. 66), the existence a n d uniqueness of the positive n t h root of a (p. 123), and the existence of the m a x i m u m of a certain product of numbers (p. 146). Such subtle issues are almost never raised in high school mathematics. Those who believe thar mathematics in the schools should only be strands around real-world applications w o u l d no doubt dismiss any discussion of such issues as sterile elitism. I k n o w otherwise, however. M y experience with teaching the basic existence and uniqueness theorem of ODE in calculus courses tells me that most of the students never get it, no matter h o w hard one tries. If the students were given ample exposure to these ideas in high school, would they not be more likely to develop this kind of mental agility? A final class of examples has to do with the book's effort to present more than one solution to a given problem whenever possible: for example, the solutions to problems on p. 36, p. 68, p. 75, p. 76, p. 84, p. 85, p. 117, a n d p. 118; three ways to s u m a geometric progression; three proofs of the inequality of arithmetic and geometric means; and so on. A great deal of the current reform effort seems to be put on convincing students that there is more than one correct answer to a problem and that there is more than one w a y to do a problem. The former is clearly a very dangerous position that, in the hands of someone less than completely knowledgeable, can lead to frightful abuses. A n d it does, see Ref. 8. The latter is m u c h discussed in the reform movement, but the discussion often lacks substance because the examples used are usually trivial. Algebra can serve as a good model to show, correctly for once, that mathematics is, indeed, "open-middled" and "nonrigid." With all these good things going for it, w o u l d it not be natural to propose Algebra as a text in the schools? I believe several changes have to be made before it is suitable as a textbook. Some of them are trivial, others m a y be less so. Let us go through them systematically: (1) The book needs an index. (2) The book needs more exercises. The problems (some of them with solutions) scattered throughout all three volumes are educational, interesting, at times amusing, and always stimulating. 9 However, a textbook needs some easy exercises for the weaker students, and it also needs a few more than what are presently in Algebra to keep the stronger students busy. More specifically, a textbook on algebra w o u l d need plenty of w o r d problems to force the students to learn to read and to translate the verbal information into mathematics. A n y o n e w h o has ever taught at the elementary level w o u l d understand at once that the latter "translation" process is the w e a k spot in most students' mathematical armor and that addressing this weakness must be one of the main concerns of an algebra course. 9 The problem on p. 3 needs s o m e tightening of the language. As it stands, 8 + - - - + 8 (125 times) is obviously a solution, but the Solution on that page conveys the false impression that the solution is unique.
(3) Some topics need to be added. Complex numbers, a thorough discussion of the roots of cubic polynomials with real coefficients, the concept of a function together with an elementary discussion of the exponential and logarithmic functions, are obvious items in an algebra course that are presently missing in Algebra. Actually, much can be gained in this book if the function concept is introduced and seriously discussed. Take the present treatment of polynomials and rational expressions, for example. Section 29 on p. 47 tries to say that two polynomials can be equal in two different senses, as members of I~[x] or as functions on ~; but it really does not get the point across too well because the function concept is missing. The same remark applies to the equality of rational expression in w (pp. 56 -58). There the situation is even more critical than the case of polynomials because the need for the concept of the d o m a i n of a function becomes acute. There is a good additional reason for introducing the function concept in algebra: w h e n students really get to k n o w polynomial functions, their entry into the world of calculus w~i1 be that m u c h smoother. (4) The informality of the exposition has to be reined in. This recommendation clearly needs a lot of'explanation! It has been said that every author of a successful textbook has to be something of a pedant. This is because, for a textbook to be of service to ALL students (so one tries, at least), the i's have to be dotted and the t's crossed. Now, Algebra maintains the same conversational style as in FG and MC and is very charming. In fact, the first paragraph of the book says clearly, "This book is about algebra. This is a very old science and its gems lost their charm for us through everyday use. We have tried in this book to refresh them for you." But there comes a time w h e n charm has to give w a y to official business, and mathematical clarity must precede all other considerations. I will illustrate this point with several examples. (4a) On p. 10, the book tries to explain that a finite product of integers is independent of the order the multiplication is carried out. As this comes right after the associative law of multiplication, the book probably expects the reader to know that associativity is involved. Nevertheless, a textbook to be used by all types of students should mention the associative law, and this is not done. (4b) On p. 29, a n is defined for n E Z, but nowhere does it say a ~ 0 for n = 0. (4c) The definition of a polynomial on p. 44 says it is ... an expression containing letters (called variables), numbers, addition, subtraction and multiplication. Here are some e x a m p l e s m : a 4 + a3b + ab 3 q- b4, (5 - 7 x ) ( x - 1)(x - 3) +
11, 0, (x - y)100.These examples contain only addition, subtraction and multiplication, but also positive integer constants as powers. It is legal because they can be considered as shortcuts (e.g., a 4 may be considered as short notation for
10 I omit more than half of the examples actually listed in the book. THE MATHEMATICALINTELLIGENCERVOL.17,NO. 1,1995 73
a 9a . a 9a which is perfectly legal). But polynomials.
a -7
or
x u are not
M y guess is that the b o o k tries to define a polynomial as an element in the ring generated by monomials, but not having the language to do so, it tries to compensate b y talking about it in an informal way. In this case, the informality is not a help and a more formal definition w o u l d probably fare better. As a matter of fact, w h e n the book comes to fractions of polynomials on p. 63, it feels necessary to add: When we say a polynomial must not contain division it does not mean that all its coefficients must be integers; they may be any numbers, including fractions, so for example, 89is a perfectly legal polynomial of degree zero. This most likely increases the confusion rather than clarifies it. (4d) I have already mentioned that the discussion of the equality of two polynomials on p. 47 is hobbled by the absence of the function concept. As it stands, I am not sure that the informal discussion there conveys the intended message at all. (4e) On p. 49, the p r o b l e m is posed: Is it possible when multiplying two polynomials that after collecting similar terms all terms vanish (have zero coefficients)? Then it goes on: Answer. No.
Remark. Probably this problem seems silly; it is clear that it cannot happen. If it is not clear, please reconsider the problem several years later. One can sympathize w i t h the authors for not wishing to o p e n a can of w o r m s [cf. (4d)], but I doubt that a student w o u l d find the preceding passage either informative or edifyingfl (4f) On p. 56, w a rational expression is defined. Again, the book probably wishes to say that a rational expression is an element of the quotient field of ~[x], but it chooses to do so informally. The resulting exposition, too long to quote here for a change, is quite a bit less than clear. (4g) On p. 58, still on the subject of rational expressions, w e find: Strictly speaking, the cancellation of common factors is not a perfectly legal operation, because sometimes the factor being cancelled may be equal to zero. For example, (x 3 + x 2 + x + 1) / ( X 2 - - 1) is undefined when x = - 1 (but) (x 2+ 1) / ( x - 1) is defined. Usually this effect is ignored but sometimes it may become important. This is charming for a mathematician to read but may v e r y well be a nightmare for a beginner. 11 It is n o t i m p o r t a n t , b u t n o t e t h a t the a n s w e r s h o u l d n o t b e " n o " b e c a u s e o n e of t h e factors c o u l d b e (x - x).
74 THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 1, 1995
(4h) Section 61 on pp. 120-121 deals with the solution of a special fourth-degree equation by reducing it to solving a quadratic equation through a clever change of variable. It is very nice, but one can give the students a better perspective if one adds a sentence to this effect: "If one is lucky, one can solve a few equations of high degree b y a special substitution." (4i) On p. 126, the concept of fractional powers of a n u m b e r a is defined without mentioning that a must be positive. There are other examples of this type, but I think I have made m y point about the need for pedantry. (The curious reader m a y wish to check the following: p. 72, lines 9-10; p. 86, lines 7-8; p. 98, last paragraph; p. 100, paragraph above w p. 117, first solution to the first problem.) (5) There should be more applications in the book. By applications, I do not mean exclusively real-world applications. Applications to other parts of mathematics which involve enlightening ideas or techniques, such as the counting of lattice points inside a circle on p. 48 of MC, w o u l d be perfectly legitimate. All the same, I think a textbook in school mathematics should have as m a n y applications as possible, for at least two reasons. First, students need applications to d e e p e n their understanding of both the new concepts and the new techniques. Second, high school students deserve to be given a wellr o u n d e d v i e w of the subject in o r d e r to develop their o w n interests or to make the correct career decisions. Peter Hilton p u t it very well w h e n he wrote (Foreword to Ref. 13): We must certainly take into consideration the potential users of mathematics, since the main argument for the importance of mathematics today is precisely the ubiquity of its many applications . . . . We also have a responsibility, as teachers of mathematics, to cater to the future citizen, the future adult. For the case at hand, a good part of algebra was created in response to needs in the other branches of mathematics as well as in the real world, and the students should be informed of this fact, including the connection with the real world, through well-chosen materials on applications. The addition of problems of an applied nature plus a few supplementary sections on applications w o u l d bring about a more balanced presentation. One last suggestion I can m a k e is to flesh out the exposition of Algebra. Its Spartan character, everything said only once and not a word to waste, m a y be intimidating to some students at this level. A few additional reminders or back references would certainly be welcome. In discussing the equality of two rational expressions on p. 72, for example, w o u l d it not help the students if a reference were m a d e to p. 47 concerning a similar problem with polynomials? Again, w h e n the existence and uniqueness of the square root of c is discussed on p. 98, a reference to p. 54 w h e r e this issue first comes u p w o u l d refresh the student's memory.
A Concluding Thought W h a t we have here are three excellent mathematical w o r k s from which students and teachers alike w o u l d have much to learn. Yet they do not fall within the prescriptions of the prevailing trends in mathematics education reform. I have tried to emphasize the qualities that m a k e these books stand out. To all of us w h o still have the goal of teaching MATHEMATICS, these books have something special to often It is a pity that m a n y in the reform m o v e m e n t choose to close their eyes to the merits of these books, the m o r e so because some h a v e even dismissed them on the g r o u n d that they do not conform to the NCTM Standards. The reform is much the worse for that.
Emeryville, CA 94608.
8. H. Wu, The rOle of open-ended problems in mathematics education, J. Math. Behav. 13 (1994), 115-128. 9. Sally M. Reis, How schools are shortchanging the gifted, Technol. Rev. (April 1994), 38-45. 10. National Excellence: A Case For Developing America's Talent, Washington, DC: U.S. Department of Education (1993). (Accessible on the computer through gopher forum.swarthmore.edu.) 11. D. Rosa, Letter to the editor, Notices Amer. Math. Soc. 41 (1994), 3. 12. John Saxon, The coming disaster in science education in America, Notices Amer. Math. Soc. 41 (1994), 103-105. 13. College Mathematics: Suggestions on How to Teach it, Washington, DC: Mathematics Association of America (1979). Department of Mathematics University of California Berkeley, CA 94720 USA
Acknowledgments The comments of C. de Boor, M. Bridger, M. Protter, C.H. Sah, and especially R. Stanley on a preliminary version of this review led to significant improvements. I wish to thank them warmly.
References 1. Curriculum and Evaluation Standards for SchoolMathematics, Reston, VA: National Council of Teachers of Math (1989). 2. Interactive Mathematics Program, Mathematics Curriculum for 9-12, ongoing. ,3. The North Carolina School of Science and Mathematics, Contemporary Precalculus Through Applications, Dedham, MA: Janson Publications (1992). 4. D. Hughes-Hallet et al., Calculus, New York: Wiley (1994). 5. ARISE, Information material available (1993) from COMAP, Inc., Suite #210, 57 Bedford Street, Lexington, MA 02173. 6. M. Serra, Discovering Geometry, San Francisco: Key Curriculum Press (1989). 7. Interactive Mathematics Program information material, available (1993) from IMP National Office, 6400 Hollis St. #5,
Gray, continued from p. 67 References 1. E. T. Whittaker, A History of the Theories of Aether and Electricity, vol. 2, London: Nelson (1953). Dover reprint of both volumes bound as one, 1989. (Miller's preface appears in the Tomash, New York reprint of 1987.) 2. J. Leveugle, Henri Poincar6 (1873) et la relativitG La Jaune et la Rouge No. 494 (1994), 29-51. 3. H.PoincarG La science et l'hypothOse, (J. Vuillemin, ed.),Paris: Flammarion (1902); reprint 1968. 4. A. I. Miller, A pr6cis of Edmund Whittaker's "The Relativity Theory of Poincar6 and Lorentz," Arch. Int. Hist. Sci. 37 (1987), 93-103. 5. H. Poincar6, La valeur de la science, Paris: Flammarion (1905). 6. H. PoincarO, Science et Mdthode, Paris: Flammarion (1908), 226-227. 7. A.I. Miller, Albert Einstein's Special Theory of Relativity, Reading, MA: Addison-Wesley (1981), 225. Faculty of Mathematics and Computing Open University Milton Keynes, MK7 6AA, England.
STATEMENT OF OWNERSHIP, MANAGEMENT, AND CIRCULATION(Required by 39 U.S.C. 3685). (1) Publication rifle: Mathematical InteUigencer. (2) Publication No. 001-656. (3) Filing Date: 10/1/94. (4) Issue Frequency: Quarterly. (5) No. of Issues Published Annually: 4. (6) Annual Subscription Price: $39.00. (7) Complete Mailing Address of Publication: 175 Fifth Avenue, New York, NY 10010. (8) Complete Mailing Address of Headquarters or General Business Office of Publisher: 175 Fifth Avenue, New York, NY 10010. (9) Full Names and Complete Mailing Addresses of Publisher, Editor, and Managing Editor: Publisher: Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010. Editor: Chandler Davis, Department of Mathematics, University of Toronto, Toronto, M5S IA1, Canada. Managing Editor: Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010. (10) Owner: Springer-Verlag Export GmbH, Tiergartenstrasse 17, D-69121 Heidelberg, Germany, and Springer-Verlag GmbH & Co. KG, Heidelberger Platz 3, D-14197 Bedin, Germany. (11) Known Bondholders, Mortgagees, and Other Security Holders Owning or Holding 1 Percent or More of Total Amount of Bonds, Mortgages, or Other Securities: Dr. Konrad Springer, Heildelberger Platz 3, D-14197 Berlin, Germany. (12) The purpose, function, and nonprofit status of this organization and the exempt status for federal income tax purposes: Has Not Changed During Preceding 12 Months. (13) Publication Name: Mathematical Intelligencer. (14) Issue Date for Circulation Data Below: 9/29/94. (15) Extent and Nature of Circulation: (a.) Total No. Copies (Net Press Run): Average No. Copies Each Issue During Preceding 12 Months, 8903; Actual No. Copies of Single Issue Published Nearest to Filing Date, 8900. (b.) Paid and/or Requested Circulation: (1) Sales Through Dealers and Carriers, Street Vendors, and Counter Sales: Average No. Copies Each Issue During Preceding 12 Months, 937; Actual No. Copies of Single Issue Published Nearest to Filing Date, 903. (2) Paid or Requested Mail Subscriptions: Average No. Copies Each Issue During Preceding 12 Months, 4697; Actual No. Copies of Single Issue Published Nearest to Filing Date, 4704. (c.) Total Paid and/or Requested Circulation: Average No. Copies Each Issue During Preceding 12 Months, 5634; Actual No. Copies of Single Issue Published Nearest to Filing Date, 5607. (d.) Free Distribution by Mail: Average No. Copies Each Issue During Preceding 12 Months, 415; Actual No. Copies of Single Issue Published Nearest to Filing Date, 119. (e.) Free Distribution Outside the Mail: Average No. Copies Each Issue During Preceding 12 Months, 260; Actual No. Copies of Single Issue Published Nearest to Filing Date, 5,35. (f.) Total Free Distribution: Average No. Copies Each Issue During Preceding 12 Months, 675; Actual No. Copies of Single Issue Published Nearest to Filing Date, 654. (g.) Total Distribution: Average No. Copies Each Issue During Preceding 12 Months, 6309; Actual No. Copies of Single Issue Published Nearest to Filing Date, 6261. (h.) Copies Not Distributed: (1) Office Use, Leftovers, Spoiled: Average No. Copies Each Issue During Preceding 12 Months, 1658; Actual No. Copies of Single Issue Published Nearest to Filing Date, 1737. (2) Return from News Agents: Average No. Copies Each Issue During Preceding 12 Months, 936; Actual No. Copies of Single Issue Published Nearest to Filing Date, 902 (i.) Total: Average No. Copies Each Issue During Preceding 12 Months, 8903; Actual No. Copies of Single Issue Published Nearest to Filing Date, 8900. Percent Paid and/or Requested Circulation: Average No. Copies Each Issue During Preceding 12 Months, 89.30%; Actual No. Copies of Single Issue Published Nearest to Filing Date, 89.54%. (16) This Statement of Ownership will be printed in the Vol. 17 #1 issue of this publication. I certify that all information furnished on this form is true and complete.
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THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 1,1995
75
Robin Wilson*
Renaissance Mathematics Textbooks By the end of the 15th century the Renaissance, a period of intellectual reawakening throughout Europe, was well under way. One cause of this reawakening was the development of the printing press, which made scholarly works much more available than previously. In consequence, there appeared a rash of new textbooks, particularly in mathematics. Among the most well known of these was the Summa of the Italian friar Luca Pacioli (c. 1445-1517), published in 1494 and commemorated recently on an Italian stamp; this work was a compilation of results in arithmetic, algebra, geometry and double-entry bookkeeping. Other influential texts included those of A d a m Riese (1492-1559), who has been commemorated on two German stamps, issued
in 1959 and 1992. Riese was the most celebrated of the German Rechenmeisters ("reckon-masters" or arithmeticians), who taught mathematical skills outside the universities. Many of the textbooks produced at this time were arithmetic texts of a rather computational nature, that helped to establish the Hindu-Arabic numerals in Europe. Another important consequence of the appearance of such books was that terminology and notation became increasingly standardized. In particular, it was around this time that the now-familiar symbols + and - started to be widely used; the symbols x and - were, however, not in widespread use until somewhat later.
Adam Riese Luca Pacioli
Arithmetic Symbols
Adam Riese
Grateful though we are for the wealth of philatelic marvels Robin Wilson has provided over the years, we need not condemn to silence all the other readers who have prizes of their own to share. Feel free to submit guest columns! They may present stamps, or coins, or for that matter bills, having some association with mathematics. Submissions may be sent to the Column Editor or to the Editorin-Chief.
Invention of Printing
*Column editor's address: Facultyof Mathematics,The Open University,MiltonKeynes,MK76AA,England.
Chandler Davis
