N-dimensional variations on themes of Pythagoras, Euclid and Archimedes

LENY > N. JEAN-MARC LEVY-LEBLOND 2S} mensional Variations n Themes of >vthagoras, UC id, and Archimedes odern culture is characterized by a lively interest in its past. This continual going back lo sources is made possible by a radically new situation: for the first time in its history, leumanily can have access to most of what il has produced. Current technologies permil the literary, musical, and artistie creations of all times to be reproduced and distributed at low cost, so that everyone-—at least in the prosperous parts of (he world—has available the whole of the human cultural heritage. There is surely another reason for our archeophilia, namely, the weakening of common cul- These ideas may be evident as regards art, but they are not so clear when applied to seience, looked at in its relation (or lack thereof) with culture. Indeed science, at least since the beginning of the twenty-first century, offen vaunts its absolute modernity and demands a radical contemporancousness, even an CH sential amnesia, relegating wide. Feeling great uncertainty about the future, one Modern advances can give all interest in the past to the us unexpected insight INTO — stuus of an optional em bellishment. Scientists toold results, casting light on day show a lack of historinaturally turns to the past to find inspiration and dieal culture unmatehed in any other intellectual protural values under the profound historic changes we are undergoing — planetrecent developments. rection, or just comfort. The Renaissance is the archtype of such a backward look yielding progress. Without a past we lose the future. Hence the importance of regular revisiting of masterpieces: Euripides and Shakespeare, Cervantes and Hugo, Monteverdi and Schubert, Giotto and Delacroix will always help us to live, love—and die. Provided, that is, that these great works are properly re-created (reinterpreted), not in the vain attempt of finding their original meaning, but rather seeking to get new meanings from them. We must listen, read, and look at what comes from the past with the present's ears, eyes, and minds; Bach cannot be the same after Stravinsky. or Titian after Picasso. fession. To be sure, the most ereative minds may feet an netive interaction with their forerunners, and many of the greal advances of the last century show the mark of an explicit dialogue with the past. Einstein was perfectly aware of confronting Galileo and Newton; and more particularly Abraham Robinson, in developing non-standard analysis, explicitly related it 10 Leibniz. But at the ordinary level of teaching, popular writing, and even research, such ties to the past are unusual. We may teach and publish on history of science, but orti narily without connection to actual scientific practice. This is too bad. Modern advances can give us unexpected insight into old results, casting light on recent developments,

JEAN-MARC LEVY-LEBLOND -Dimensional Variations |Nemes of Pytnagoras, id, and Archimedes odern culture is characterized by a lively interest in its past. This continual going back to sources is made possible by a radically new situation: for the first time in its history, humanity can have access to most of what it has produced. Current technologies permit the literary, musical, and artistic creations of all times to one—at least in the prosperous parts of the world—has These ideas may be evident as regards art, but they are not so clear when applied to science, looked at in its relation (or lack thereof) with culture. Indeed science, at least available the whole of the human cultural heritage. There is surely another reason for our archeophilia, namely, the since the beginning of the twenty-first century, often vaunts its absolute modernity and demands a radical contempobe reproduced and distributed at low cost, so that everyweakening of common cultural values under the profound historic changes we are undergoing planetwide. Feeling great uncertainty about the future, one naturally turns to the past to find inspiration and di- Modern advances can give us unexpected insight into old results, casting light on recent developments. raneousness, even an essential amnesia, relegating all interest in the past to the status of an optional embellishment. Scientists today show a lack of historical culture unmatched in any other intellectual prorection, or just comfort. The Renaissance is the archtype of such a backward fession. To be sure, the most creative minds may feel an active interaction with look yielding progress. Without a past we lose the future. Hence the importance of regular revisiting of masterpieces: Euripides and Shakespeare, Cervantes and Hugo, Monteverdi and Schubert, Giotto and Delacroix will always help their forerunners, and many of the great advances of the us to live, love—and die. Provided, that is, that these great works are properly re-created (reinterpreted), not in the vain attempt of finding their original meaning, but rather last century show the mark of an explicit dialogue with the past. Einstein was perfectly aware of confronting Galileo and Newton; and more particularly Abraham Robinson, in developing non-standard analysis, explicitly related it to Leibniz. But at the ordinary level of teaching, popular writing, and even research, such ties to the past are unusual. We may teach and publish on history of science, but ordiseeking to get new meanings from them. We must listen, read, and look at what comes from the past with the presnarily without connection to actual scientific practice. This ent’s ears, eyes, and minds; Bach cannot be the same after Stravinsky, or Titian after Picasso. is too bad. Modern advances can give us unexpected insight into old results, casting light on recent developments,

Figure 1. Py thagoras in B the plane, Just as à mo dern Produc tion of Anti reveal new gone or Ki Meanings an ng Lear ca d have curr n This may be ent impact, 100 Pompou s an introduc amples I wa tion forà fe nt to offer, w ex. They are dr est fields in awn from on the World, SC e of th e old. Ometry— un Cist's Figure 2, Py thagoras in Sp ace, sense, as “m derstood in the Physieasure Of sp ace.” Some ancient) resu Classical (eve lts have inte n resting Sene Mensions, | ra li za think they s tions to N how the Perm diditional theo anence of th rems and also (Area ABC)? ese traMay help in tuition for hi forming a be gher Space = (Area OAB) tt er indimensj Ons. I have been ? + (Area OB The Collecti complaining C)? + (Area ve am ne sia about makes OCA)? much origin A fairly easy it hard to kn (2) ality to clai pr oo f o Co w ul how m for the pr d be gi fo mathematicia ve rmula for th n es St en ar t ti re ng sults. That from Heron' e area of an n and physic most s arbitrary tr ist colleagu lengths 4, b, to give me iangle in te es had no re e indicates Mo of it rms of the s si fe de rences s: stly that ¢ T found by go hey Shared od luek that the amnesia, one of the re Ata + (a first), was sults anyway Fr published in a aye (t he a lo ng Seometer H. -ago article + a — a S, M. Coxete by + p — €) th e gr ea r. t I will be gratef ther inform ation on th ul fo " r g an p 20202 — gi y fure questions + ci take n up here, rcular perm,) , (3) Getting the Pythagoras Sides of the and the Or triangle ABC thosimpiex its Projection Let us begi in terms of in N Dimens n by tevisiti s by the ( us those of io ns ual) Pythagor ng th e Pythagoras, ve ry ean theorem, an ci ent theorem and Senerali b j ce? hie c zing it ta ar of Course, Lam z a, Au bitrary dime not talking ab ge +2 nsion, Of ou (4) t Wi the metric Ve ROW taken as th th e no ta tions of Figu rsion, which an axiom in re is 2, de fi Pu of a resnÿt wh ning euclidea tting this Ba ck into eq. (3 n Spaces, bu ich May be ), pleasant if no t The classica t pr A° = ju p ofound, ! l result will a r fi p rs e t be Presente Which is to pie tea be Seneralize d in the form d. Let twa li which is Ju orthogonally ne s st in the plane the result an at O. For an meet nounced in y segment A (Fig. 1), the Actually, as eq. (2), B Joining th Square of its aften happen es e li ne s, s le giving a more ng en th of the length ab is le th s e us s u to appropriate m of the squa see the genera s of the two Proof res lity of the re Segments O the area of tr Cuts off on th sult. We can A and OB, ia e lines—its ng le A wh express BC a8 a vect ich it Iwo orthogon wh ose length is or Orthogonal al] Projection the vector Pr to its plane s: oduct of any AB? = OA? two ofits side 4 OB? s: Naw in Eucl (1) idean 3-Spac e Consider thre orthogonally e lines Meet at O. Let AB ing C be a triang Now we Ju each of thes le with 4 Ve st notice th e lines, whos rt at ex on e Pr fermined by ojections On pairs of thes the planes de n e li e nes are then OCA (Fig, 2) BD . Then OAB, OBC, an eat d p and we see (6) th at Theorem of Pythagoras 3D . Th Of triangle A e Square of th BC equals th A= Le D e area e sum of the IA (a. areas of its ©) squares of th three Projecti e ons: 1 = à D Ae + hope ) £ Nat a g The result wa a Sb‘ (7 s, I thought, ) new I then ha rem,” Math, pp ened 10 fin Gazette 19 (1 d it in the art 935), 206 Th iti e by H, 5, M. eir result 18 jus Coxeter an t the same, d P.S Donc but their Proo hian, "An f is more Stra ft-dirmensio ightiorward naf extensio n of Pytha, ga ras's thap44 nb MA THEMATICAL INTE) LIGENDER

The three terms are the vector areas of triangles OAB, OBC, OCA, respectively. These vectors being mutually perpendicular, the usual Pythagorean theorem immediately gives the result in eq. (2). This demonstration fits into a more general interpretation. Let us define the “vectorial area” of a surface Y as the vector Ay:= | nds (8) where n is the normal to the surface. Then for every closed surface, the vectorial area is zero. The physical interpretation is simple:* for any vector u, the quantity u - Ay is the flux of the constant field u across the closed surface Y, which is zero (if it is necessary to convince oneself of this, one may transform the flux to the volume integral of the field’s divergence, which is zero). Applied to any tetrahedron, this says that the sum of the vectorial areas of the four faces is zero. If three of these faces are mutually orthogonal, as they are for the tetrahedron OABC under consideration here, this means that the vectorial area of the fourth face is the vector sum of those of the three mutually orthogonal faces; now eq. (2) follows by applying the usual 3-dimensional Pythagorean theorem. Now take N orthogonal lines meeting at O in N-dimen- À = (ro Arg A. Ary) = m Arg À... Arn). The first term here gives the volume of face F;, the N-simplex (O, Ps, P3,..., Py), as vector A, orthogonal to that face. The same applies to each of the other terms, one corresponding to each of the right faces. This gives (with appropriate choice of orientations) Exactly as in the 3-dimensional case, this is valid for every simplex. But here, starting from an orthosimplex, we have the added feature that the projections (the right faces) are orthogonal: A; is parallel to the Ath axis; the usual Pythagorean theorem now gives eq. (2). Note, however, that the generality of the result is limited: the N-simplexes, which can be defined by N points on N orthogonal lines, are, forN > 3, very special. Indeed, they are determined by specifying N parameters #1, 72, ..., TN, while the general N-hedron depends on NN — 1)/2 parameters (for example, the lengths of its edges). sional Euclidean space, and an N-simplex formed by N points Euclid and the N-Dimensional Pyramid Conceptual mastery of space goes by way of geometry, regarded in the first place as science of the measure of P;, Ps, ..., Px, one on each line. Together with O, these forms—thus fundamentally a physics of space. Estimation make an (N + 1)-simplex, all of whose face angles at O are of lengths, areas, and volumes of the simplest objects in 3space is its core.* The importance of the formulas giving the area of a triangle and the volume of a pyramid cannot be overstated; these are, after all, the first examples of integration in 2 and 3 dimensions, as was realized later. Now the geometric reasoning which establishes these two results generalizes easily to N dimensions, furnishing both a new proof of elementary results and an approach which ought to instill some N-dimensional intuition. The case of the triangle is quickly disposed of, for every triangle has evidently area half that of the rectangle with the same base and altitude (Fig, 3). But I prefer here to start from a special case which will be easily generalizable: an isosceles right triangle, half of the square having the same side. The formula giving its area, right angles, which we accordingly call an “orthosimplex.” Call its (N — 1)-face H := (Pj, Po, ..., Py), its “hypotenusal” face, and call its N faces F,:= (0, Py, Po,..., B,..., Pò k=1,2,...,N, the “right” faces; these are the projections of the hypotenusal face parallel to the axes; they too are orthosimplexes, in (N — 1) dimensions. Then Theorem of Pythagoras ND. The square of the (N — 1)-dimensional volume of the hypotenusal face of an orthosimplex equals the sum of the squares of the volumes of its N right faces: N (Voly_1 HJ? = I (Voly_1 Fy)”. (9) 1 The proof is almost trivial if one invokes the exterior calculus. Setting r; = OP, for the vector to the point P; (k = 1,2,...,.N), one may express the (N — 1)-dimensional volume of the N-simplex (Pi, Ps, ..., Py) as a vector orthogonal to the (N — 1)-dimensional subspace containing it, by taking the exterior product of (N — 1) of its edges, for example those away from the point Pi; Area of triangle = mi Altitude x Length of base (13) then extends to an arbitrary triangle by suitable affine transformations, which make the triangle skew and dilate its dimensions, retaining validity of eq. (13) and the factor ! (Fig. 4). A= (re = ri) IN (Ts = ri) ess A (ry = ri). (10) The extension to 3 dimensions is still easy. Consider a cube, and make use of the 3-fold symmetry about an axis A simple proof by recursion shows that, by antisymmetry of the exterior product, the right-hand member is equal to the sum of the terms given by the exterior product (up to joining two opposite vertices. A clever dissection into three equal pyramids joined along that main diagonal (Fig. 5) sign) of (N — 1) of the N vectors ry (k = 1,2,..., N): shows that the volume of a right pyramid with square base “Thanks to Jean-Paul Marmorat for this remark, “The notation means that the face F, contains all the points OP, Ps Py with the exception of Py. *Euclid, Elements, Book XII, VOLUME 26, NUMBER 4, 2004.

equal pieces. Indeed, any point (ri, 2, . EN) E TO (that is, one such that 0 Sa, = 1 Vk = 1,2,...,N) belongs to one and only one of the hyperpyramids, namely, that [I° such that, = sup (a7), 4a, ry). An evident consequence is that each of the hyperpyramids has volume equal to 1/N. The result generalizes to an arbitrary hyperpyramid in the same way as above, showing therefore that Volume of N-hyperpyramid | . 5 a Altitude x Volume of (N — 1)-hyperbase. Figure 3. The area of a triangle. (15) In algebraic terms, this is evidently equivalent to integrating a monomial of exponent (N — 1). Indeed, consider and with altitude equal to the side of the square is indeed a third of the volume of the cube: the section of the hyperpyramid TIyN (for example) by the plane ry=t (Q<t<1):it is an (N — 1)-hypercube whose vertices are the points (ta, ta», ..., ly, 1) Volume of pyramid = 3 Altitude x Area of base (14) with each a, = 0 or 1 (kh =1, 2,...,N— 1); its volume is {*° 1, Then the total volume of the Again, suitable affine transformations carry the pyramid into any other pyramid with parallelogram base. Finally, a Cavalieri-style comparison of such a pyramid with another having the same altitude, and any base provided it has the same area as the parallelogram (Fig. 6), completes the proof that eq. (14) holds for any pyramid.” Seeing the factor | in 2 dimensions and the factor a in 3 dimensions, one is tempted to generalize. In 4 pyramid is obtained by integrating with RO ST os | = respect to the height / of the plane of hd dimensions, the picture can still be visualized. The 4dimensional hypercube (often called a “tesseract” when one wants to sound mysterious) is easily projected down to 2 dimensions (Fig. 7). One can see that it can, in fact, be dissected into four equal hyperpyramids with (3-dimensional) cubic base, joined along a main diagonal of the hypercube. Thus the volume of each is equal to a quarter of the volume of the hypercube, and this relation generalizes to other hyper- Figure 4. The area of a triangle, again. pyramids as above. Now there is a fairly simple formal proof in arbitrary dimension N. In the euclidean space € with an orthonormal basis at the origin O, consider the hypercube FN) of unit edge whose 2 vertices are given by their coordinates a}, as... ., ay, with each ay being 0 or 1 (k = 1, 2, ..., N). Denote by Fx! the hypercubic (N — 1)-face of FM defined by setting a, = 1; these N hypercubes are the N faces of PE) which meet at the vertex O' = (1, 1,..., 1) opposite the origin O = (0, 0,... , 0). Further, consider the N hyperpyramids I) ( = 1, 2,..., N) having O as vertex and as bases the N hypercubes FN D respectively. These equal hyperpyramids, with common edge OO’, partition I into N "By shameless use of “physicist’s asoning” implicitly relying on the infinitesimal, this argument totally by es the negative solution of Hilbert's Third Problem: the impossibility (unlike the case of 2 dimensions) of dissecting two tetrahedra of the same volume inte congruent elements. Figure 5. The dissection of the cube and the volume of a pyramid.

Figure 6. The volume of an arbitrary pyramid. Figure 7. The dissection of the hypercube and the volume of a hyperpyramid.

the section. In conclusion, this shows by purely geometric methods that The second gives the expression aria fu Neerexp(—?) rl | di 1 = 0 (16) N° = | ant taining the factor 1/3 by dissection of the cube into pyramids is not exactly the method Euclid gives. According to an earlier proof of Eudoxus, whose idea is generally attributed to Democritus,” the result is obtained by trisecting a prism into =o OP an =“ T(N/2) pg9 OD _ TUOI * The double jump of 7 Among the many mysteries afforded by the number 7,’ not the least is the following: not content to relate the circumference and the area of the circle,“ the very same number appears in the formulas for the area of the sphere and its volume.” But better yet, the expressions for area and volume of the N-dimensional sphere get along without requiring any other transcendental number—as they perfectly well might, after all, at least before one has done any calculations. These formulas all appeal only to 7, raised to powers which jump in every other dimensionality, thus generalizing the behavior observed in dimensions 2 and 3. Table 1 displays this strange phenomenon:!" 3 An | 0 | 2 | 278 | 478 Wy 4 | 5 6 (ky DATA T1+N2) da Now let us make this more explicit, distinguishing according to whether the dimensionality is even or odd. In the former case, N = 2p, the denominator is a factorial of an integer, and in the numerator there are N factors V7, so p factors 7. In the latter case, a factor V 7 comes into the denominator which takes care of one of the (2p + 1) factors V 7 in the numerator, finally leaving only p factors 7. This compensation, which in this account seems perfectly accidental, gives rise to the “double jump” of 7, without giving any understanding of its geometric necessity. I would like to present now two other methods of calculation which clear up the mystery of the double jump of powers of 7; one of them is based on a simple and elegant recursion which has the virtue of generalizing some results more than two millennia old. In 3 dimensions (Archimedes) Table 1 Area and Volume of the N-Dimensional Sphere |0|1 | 2 DN “ NT(N2) also generalizes to N dimensions. Archimedes and the N-Dimensional Sphere (18) a This yields the final result (for the volume, one integrates the surface of the sphere of radius r, from 0 to R): three pyramids (not necessarily equal) of the same volume. Leave to the reader the pleasure of verifying that this method N dr exp(—r?) = Jat W2)R®. 0 The calculation of the volume of the pyramid and ob- 7 To begin with, recall the classical calculation of the area of ZR the ordinary sphere (the sphere in owr space). Archimedes was the first to show!! that the area of a sphere is exactly [1 | 2A | wR | gaf | SR equal to the lateral area of the circumscribed cylinder of the same radius R and with altitude equal to the sphere's The most-used explicit calculation throws little light on these results. The classical “dodge” for calculating the area of the sphere in N dimensions consists of integrating over Euclidean space the Gaussian function, which combines the properties of factoring into functions (Gaussian) of a single variable and of having spherical symmetry. The first of these properties allows one to write The simplest proof consists in considering on the sphere of radius À the parallel of latitude determined by the polar angle 6. The infinitesimal zone between this parallel and another distant dl from it (on the sphere) has area dA = 2rR sin # dl, because the small circle at that latitude has radius + = R sin 4. But the parallel planes containing the two circles cut off on the cylinder circumscribed at the equator a band of altitude dh = sin 9 dl. The area of the | = dNr exp(- 12) = | 1. [ dix, diva... dry exp(—a7 . . —a®) N = [ae exp(—a")| R diameter (Fig. 8). =[T(/2))N= N2RN band is therefore dA’ = 27R dh = dA, equal to that of the infinitesimal spherical zone. The sphere and the cylinder thus have the same total area: (17) A = 2aRH = Ank. (20) | "See the long note of Thomas L. Heath in The Thirteen Books of Euclid's-Elernents, Dover, New York, vol, Ill, pp. 365-368. See Jean-Paul Delahaye, Le fascinant nombre Pi, Belin, 1999 I know, | know: | said it that way when | was a child, but nowadays we are supposed to say “the area of the disk.” Fashions of the times. “Beg pardon, The volume of the ball. seeing that the circumference of a circle is 277 while the surface of a sphere is 474°, we might be tempted to expect the hypersurface of a hypersphere [in 4 dimensions] to be 678° or 8785. It is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression, 27°R°.” H, S. M. Coxeter, Regular Polytopes, Macmillan, 1963, p. 119. In his course on Mathematical Methods of Physics in the 1960s, Laurent Schwartz scored a succe ss with his expressions of regret at not living In a space of 6 dimensions, where the formula for surface area Is "so beautiful." “Archimedes, Oeuvres complètes, vol. 1: THE MATHEMATICAL INTELLIGENCER La sphere et le cylindre, ed, Ch, Mugler, Les Belles Lettres, Paris, 1970

Vy(R) = | AmMdr=| o . 1 . an dr = ay we “0 (23) i whence H=2R vy = Lay, VOD = LR A(R). (24) Now we can get to the heart of the argument, establishing a generalized Archimedean cylindrical projection. To this end, consider the variable point 7 = (ay, a, . .., ay) € EN, and introduce polar coordinates for its first two coordinates: Y T] = pcos w x = p Sin @ Figure 8. The area of the sphere. (73, 24, (25) AN) = 7 E EN? As to the volume of the interior of the sphere, it suffices to consider it as cut up into infinitesimal pyramids having vertex at the center and bases on the surface: for each of these In the new coordinates, the volume element becomes dNr = dajdae...dxy = p dp de dr". pyramids, of course, dV = 5 R dA, hence for the whole (26) sphere one has also V = ! RA, so Now we set up a bijection between the sphere Jy (omit- An BP: 3 ting its poles to avoid the singularity of the parametrization at p = 0) and the Cartesian product of the ball Ay (parametrized by r’, with r’ < R) with the sphere > (that V= (21) Archimedes expressed this result by declaring equal the volume of the sphere and that of the cylinder after removing from it the pyramids with vertex at the center and bases the bases of the cylinder." It is said that Archimedes was so proud of these results—rightly—that he wanted the key figure of sphere with circumseribed cylinder carved on his tombstone. Three centuries after his assassination at the siege of Syracuse, Cicero recognized his grave by this insignia; it has not been rediscovered in recent centuries, alas. The correspondence which comes in here between the sphere and the circumscribed cylinder is just the cartographer's “cylindrical parallel” (or “normal”) projection, which, we have seen as an incidental extra, preserves area locally, not only globally; but, as we knew, it distorts angles, worse and worse as one approaches the poles, where it is singular.!* is, the circle parametrized by angle ¢), in other words, a torus: (v= (ner) n= RIC Sue do E Brun <R p= VR? — ir? (27) This really does generalize the usual cylindrical projection: in the case of 3 dimensions, the sphere (3) is projected to the cylinder, which is the Cartesian product of the circle (4) and the segment (3), which we can just as well regard as a torus. In the general case too, the projection preserves measure; for the integral over the sphere fy (with its uniform measure) of an arbitrary function F, can be written in terms of the Dirac &function as | ator =| sr d\r Sr — RFC) In N dimensions IN In euclidean N-space €% let us consider the sphere fy of radius À, denoting its area by Ax(R), and its interior ball Ay of volume Va(R). Again denoting by ay and vy the area and volume of the sphere of unit radius, we have for evident reasons of homogeneity RN! ANR) = ay . n u = ) (2 Vy = Finally, simple integration just as in the 3-dimensional case, allows us to write (ee = |papagan ®r' SVI + pF — RF Cer’) (28) ~ ride | yer IN Fler), where the last equality requires the standard result that d[u(a) — u(a)] = [u'(a)] ! ar — a). Integrating here the identity function (F = 1), one obtains the area of the sphere as the product of the area of the sphere ‘fs with the volume of the ball @y>, that is, the charming formula ANCR) = 27RV,-2 (R), so that ax = 2arun-s. (29) ‘The equality in this form can be demonstrated directly by slicing the two volumes by planes perpendicular to the axis of the cylinder and observing that the two sections (a disk and an annulus respectively) have equal area. The Cavalieri theory of indivisibles (already known before the 17th century), a heunstic formi:of a rigorous theory of integration, allows one then to assert that the two solids—the sphere and the hollowed-out cylinder —made up of sections of equal areas, have the same volume Ht is commonly but mistakenly said that this projection is Mercator's, That great cartographer invented in 1569 a cylindrical projection which is conformal (preserves angles), much less trivial. It is curious, In view of its appearance in Archimedes's result, that the cylindrical normal projection was apparantly not used in cartography until the work of J. H, Lambert. See John P. Snyder, Flattening the Earth (Two Thousand Years of Map Projections), University of Chicago Press, 1993 VOLUME 28, NUMBER 3, 2004

This formula is the essential result; despite its simplicity, I have not seen it in any book.!* In view of (24), one can now give the following recurrence relations: If on the other hand the dimensionality is odd, N = 2p + 1, one makes the same change of variables as before in p planes corresponding to p pairs of coordinates, but now one coordinate is left an old maid. Thus we get, instead of eq. (34), ay = N-3 AN-2 and Uv = = Uy-», (30) Qa \P Vop+i = (7) which lead easily to the table above, © and also lead back to the formulas (19). This shows us at once why the power of 7jumps at every other step. Namely, the dimensionalities 0 and 1 are trivial and do not involve 7 at all; after that, one gets from dimension 0) to all the even dimensions, introducing an additional power of 7 each time (note that eq. (29) does hold for N = 2), and similarly one gets from dimension 1 to the odd dimensions. One 7 per plane! Here is another viewpoint on this curious difference between spaces of even dimensionality and of odd dimenp [Rr A > | [ats drop Re = kel F + te p 9 1 th — Apo 7 1 „\p a R- we ni do: ı=2 Eper L—2 du, p! JR i È p! 0 (36) or, finally, Zepel pl gb Vop4 1 — Gabi À 2p+1 37 ie in agreement with the general expressions. The preceding changes of variable can be regarded as projecting, with preservation of Euclidean measure, sionality. The volume of the sphere Yy can be written * in even dimensions, the ball Rs, onto the Cartesian product of p spheres f» (circles) with the interior of a pdimensional polyhedron; Vie= | dr OR? — 1°) * in odd dimensions, the ball %»,,, onto the Cartesian EN = [dde deeg OR? = af - a3... 2%), GD product of p spheres fs (circles) as before with the interior of a segment of a (p + 1)-dimensional paraboloid Pp (given by > le + pr = R20 where # is the Heaviside function. First look at even dimensionality N = 2p. Change variables to polar coordinates in each of the p planes defined by pairs of coordinate axes: Var = COS, oe = prsin gg (kK =1,2,...,p). (k =1,2,..., p). dependent ways to turn around in space, that is, simply the number of independent planes. That is evidently the integral part of half the dimensionality.'" À pi \ ] Dm rational, so there are as many factors 7 as there are circles. (33) « [or dpx dex ere = Vr) E kel In both cases, the volume of the polyhedron or paraboloid is The situation is simple in essence: there are as many factors 7 in the expression for volume (or for area) of the sphere in N dimensions as there are “independent circularities” in the space, meaning by that, the number of in- It follows at once by (31) that Vop = | = bia: P)). (32) Further, let 9 t k = Pk 2% k=1 D p = kel The surface effect y It may be interesting to show how the expression for the PN 1 volume of the sphere in N dimensions illuminates the phys- / and in the last integral the integrand is 1 on the p-dimensional volume interior to the pyramid described by the positive half-axes (0 < ty, k = 1, 2,...,) and the hyperplane cutting them at fj, + to + + ty. = R?, That volume is (pl)! RE, giving us finally i] Vay = — RP. (35) ical nature of high-dimensional spaces. One might call this the “surface effect” the higher the dimensionality, the more points there are close to the surface—in a sense to be clarified by the following examples. Naturally this is true for most sufficiently regular N-dimensional bodies; the only advantage of the sphere is in permitting explicit calculations leading to simple results. First let us ask this question: what is the radius of the concentric ball inside a ball of radius R which contains half Thus Coxeter (op. cit, p. 126) gives the result, in the form of the first of équations (30), but derived from the expressions obtained by the “gaussian” method, not from the geometric meaning, ‘Note in particular the case of the sphere in 4 dimensions: its area is equal to the volume of a torus in 3-space, and | wonder if the corresponding cartography would be a useful geometric tool. ‘6We may be permitted to. wonder whether the difference in behavior brought out hete between even- and odd-dimensional spaces is related to the Euler-Poincare characteristic, which also distinguishes tham. THE MATHEMATICAL INTELLIGENCER

It is a monotone increasing function, showing that the area of a sphere is larger (relative to the volume of the section) the larger the dimensionality. Lex Here is another way to look at this. Consider a uniform Kn 6. probability density on the unit ball in N dimensions. Let us ask what is the mean distance ry between a random point in the ball and the center. This is given, of course, by ‘è. 4 a. “8e. 1 “| re | dr r Ax(7) N= >, | : (40) dr Ay (ar) „0 N which becomes, by eq. (22), | 0 2 DI 6 8 10 12 14 16 (x | dr” IE “0 1 N = NE > Figure 9. The “effective size" of the unit ball in N dimensions. Lai = ni ly | dr N its volume? This requires that VCR’) = ! VR), or vyf'N = LN. i vl vR ‚or ni R'=2 INR. (38) Now plainly when N grows indefinitely, the radius R' ap- 41 2 In other words, the mean distance to the center, for large values of the dimensionality N, is very close to the radius of the ball, which means that most of the points are very close to the boundary. That means that the radius of the ball gives a misleading idea of its effective size, making it seem smaller than it proaches À. It is more striking to think instead of the outer is, because the abundance of available dimensions someshell between the inner and outer sphere, and containing half the volume of the latter. The higher the dimensionalhow compensates for the concentration near the surface. ity, the more the thickness AR = R — R' shrinks relative to the radius À, as shown in Table 2. Consider this rather surprising illustration of this interpretation, If we think that a good idea of the “effective size” of an N-ball is given by the length £y of the side of a hypercube of volume equal to that of the ball, which we may take Table 2. The Surface Effect ARR 0.5 to have unit radius, then by eq. (19) we get this length as 0.29 fy i=. (UN We see that the outer half of the volume is crowded into a thinner and thinner layer. The reason is evidently the availability of a high number—namely (N — 1)—of directions orthogonal to the radial direction, allowing a transversal “spreading,” enabling volume to be large with small thickness. In the same order of ideas, it is worthwhile to compare the surface of the sphere Yy with the volume of its “principal section,” the ball Axy-ı, by calculating the ratio Axy/Vy_1= fv/oxn-ı (it doesn't depend on the radius because of homogeneity, otherwise the quantity would have no significance). By Eq. (19), it turns out that fyloy-1= NZ, 2, = B(3 "= (2) [FC + NEN 42 aid which turns out to be a decreasing function of N (Fig. 9). Notice the obvious special cases {; = 2 (in 1 dimension, the ball of unit radius is the same as the cube of side 2) and (> = \ Tr. Equation (42) is of course not defined in dimension zero. However, its limit as N — 0 is well defined: Co = Vie’ = 2.3656..., (43) where y is Euler's constant. This apparently obliges us to regard the curious constant (eq. 43) as the effective size of the 0-dimensional unit ball (2). (39) an Eulerian function, whose first few values are given in Table 3. vs == ___ PR) _ Dimensionality and Orthogonality l close with some elementary geometric considerations on N-dimensional euclidean space € y, which will bring out the great leeway afforded by letting the dimensionality be high. A simple, naïve statement would be that in a space of high Table 3. The Surface Effect (continued) dimensionality vectors tend to be “more and more independent” (which is natural) and even “more and more orfn V1 thogonal" (which is less so). The point is that the direction 2 | # | 4 37/2 cosines of any direction (with respect to a system of or- VOLUME 26. NUMBER 3, 2004

Figure 10. The equiaxial angle. Figure 11. The isogonal angle. thogonal axes), having to add to unity, must generically all be small, making the angle between them close to a right angle. Let us check this in some special cases. NE] Nu = 0. (45) Ì Projecting this relation onto any one of the vectors, one Equiaxiality Consider a system of N orthogonal axes in Ey, and call gets the relation defining the isogonal angle, namely 1 “equiaxes” referred to them the 2\_! lines making equal ancos By = — N (46) gles with all of them; in 2 dimensions, the equiaxes are the two bisectors of the angle between the axes. Denote by av the (acute) angle between an equiaxis and a coordinate axis; call it the “equiaxial angle” (Fig. 10). Thus the unit vector along an equiaxis projects onto each coordinate axis to a segment of + cos ay so that the angle ay is given by N cos ay = 1, or These values may be tabulated as shown in Table 5 Table 5. The Isogonal Angle!” N | 1 2 | 3 Il COS AN = =. (44) 4 5 ita | WEST Bry | 180° | 120° | 109.4 | 104.5" | 101.5 1 Again, the increase of dimensionality brings with it a tendency toward orthogonality. The numerical values of the equiaxial angle for some low dimensionalities are given in Table 4. Indeed, the equiaxes are closer and closer to orthogonality to the coordinate axes. Uniformity Given the sphere 4 y in N dimensions, what is the mean angle between two vectors taken randomly on the sphere, given its uniform measure? A simple symmetry argument Table 4. The Equiaxial Angle zi a nola shows that it is a right angle. But the probability distribution of this angle 4, call it py(0), rewards attention. Let us parametrize the sphere in generalized spherical coordinates: Isogonality Just as natural is the figure which we may call “isogonal” formed in €‘ by N + 1 half-lines making equal angles with each other. In 2 dimensions this gives the Mercedes trigon: in 3 dimensions, the directions to the vertices of a regular x = cost), ito = Sind, cos®, ay = sind, sinds costs, „Ay .. . = Sind, sinds... cosfy-1, TN sindsinds ... sindx- 1. (47) In these coordinates the uniform measure on the sphere tetrahedron from its center—in general, the directions of the vertices of a regular simplex seen from its center. Let us denote by By the ‘isogonal angle” between any two of these lines (Fig. 11). By symmetry, the N + 1 unit vectors appears as u; (k = 1,2,..., N+ 1) on isogonal rays are linearly de- Of course, integration on all of the angles would give pendent by back the area of Jy calculated above. Here we are look- Note that the familiar relation Hz = 24. which dete mines several fe dimensional case, The equation Arccos (N!) = 2 4 in dN lo = dé dés... dOy THE MATHEMATICAL INTELLIGENCE (48) of spatial symmetries in our world (for example in erystallagraphy), Is altogether confined to the 3only solution is N = 3, must be adde that N = NIN — 1/2 (whenee the exterior product of two vectors can be a “vector product”) and the fact that N + 1 = 2 52 sin’ 26, sin’ 305... Sindy >, 1er particularities of our space, along with the fact | (whence two equiaxial lines can also be isagonal)

N= 100 N=10 N=4 N=3 1+ N=2 wa Jf 0 NN 0,57 8 I JEAN-MARC LEVY-LEBLOND Figure 12. The probability density of the angular distance of two ran- Physique Theorique dom points on an N-dimensional sphere, Université de Nice Sophie-Antipolis Parc Valrose ing at the angle between a randomly chosen variable di- 06108 Nice Cedex rection and a fixed direction. Let us take the zenith axis France Or, as reference direction; so it will be the random varie-mail: imil@unice.fr able #,, which is at issue. Now integrating over all the other angular variables, we see that the desired probability density, which determines the random distribution Jean-Marc Lévy-Leblond describes himself as a theoretical physicist and an experimental epistemologist, but also an esof the angular distance between two arbitrary directions, is given by a “science critic.” The review he co-founded, Alfage, has been (49) endeavoring to blend sciences and humanities for more than py(8) = K sin\ * 6. This is a distribution with mean value 4 = 7/2, more and more “sharply peaked” as the dimensionality N grows (Fig. 12); for large values of N its width is 60 = O(N 12), and it sayist and editor on the cultural aspects of science—in short, a dozen years. He has recently published a collection of offtrail chronicles on science, Impasciences, Seuil, 2008. approaches the Dirac distribution 6(4— 7/2) in the limit That is to say, the higher the dimensionality of a Euclidean space, the more likely arbitrary directions are to be close to orthogonal. This idea may have some effect on the intuition we can have (or can't have!) about an infinite-dimensional space like the Hilbert space of quantum mechanics.!* fron Ant Colony Th. Optimization Marco Dorigo and Thomas Stützle ÎV fel lt is my pleasure to € extend my deep thanks to Chandler Davis for translatin g this article into English “This is essential reading not only for those working in artificial intelligence and optimization, but for all of us who find the interface between biology and technology fascinating.” — lain D. Couzin, Princeton University, and University of Oxford A Bradford Book 328 pp., 72 Illus. $40 now in paperback Abel’s Proof An Essay on the Sources and Meaning of Mathematical Unsolvability Peter Pesic “Readers of Pesic's fascinating little book will be led to an inescapable verdict: Niels Abel was guilty of ingenuity in the fifth degree.” — William Dunham, Muhlenberg College, author of Journey through Genius 224 pp.. 46 illus, $14.95 paper To order call 800-405-1619 VOLUME NUMBER

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