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Artful Mathematics: TheHeritage of M. C. EscherCelebrating Mathematics Awareness Month

446 NOTICES OF THE AMS VOLUME 50, NUMBER 4

In recognition of the 2003 Mathematics AwarenessMonth theme Mathematics and Art, this articlebrings together three different pieces about inter-sections between mathematics and the artwork of M. C. Escher. For more information about Mathe-matics Awareness Month, visit the website http://mathforum.org/mam/03/. The site contains materialsfor organizing local celebrations of MathematicsAwareness Month.

The MathematicalStructure of EschersPrint GalleryB. de Smit and H. W. Lenstra Jr.

In 1956 the Dutch graphic artist Maurits CornelisEscher (18981972) made an unusual lithographwith the title Prentententoonstelling. It shows a youngman standing in an exhibition gallery, viewing a printof a Mediterranean seaport. As his eyes follow thequayside buildings shown on the print from left toright and then down, he discovers among them thevery same gallery in which he is standing. A circularwhite patch in the middle of the lithograph contains Eschers monogram and signature.

What is the mathematics behind Prententen-toonstelling? Is there a more satisfactory way of fill-ing in the central white hole? We shall see that thelithograph can be viewed as drawn on a certain el-liptic curve over the field of complex numbers and

deduce that an idealized version of the picture re-peats itself in the middle. More precisely, it con-tains a copy of itself, rotated clockwise by157.6255960832 . . . degrees and scaled down by afactor of 22.5836845286 . . . .

Eschers MethodThe best explanation of how Prentententoonstellingwas made is found in The Magic Mirror of M. C. Es-cher by Bruno Ernst [1], from which the followingquotations and all illustrations in this section aretaken. Escher started from the idea that itmustbe possible to make an annular bulge, acyclic expansionwithout beginning or end. Therealization of this idea caused him some almightyheadaches. At first, he tried to put his idea into

B. de Smit and H. W. Lenstra Jr. are at the MathematischInstituut, Universiteit Leiden, the Netherlands. H. W. Lenstraalso holds a position at the University of California, Berke-ley. Their email addresses are [email protected] and [email protected]

Figure 1. Eschers lithographPrentententoonstelling (1956).

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APRIL 2003 NOTICES OF THE AMS 447

practice using straight lines [Figure 2], but then heintuitively adopted the curved lines shown in Fig-ure [3]. In this way the original small squares couldbetter retain their square appearance.

After a number of successive improvementsEscher arrived at the grid shown in Figure 4. As onetravels from A to D, the squares making up the grid expand by a factor of 4 in each direction. As onegoes clockwise around the center, the grid folds onto itself, but expanded by a factor of 44 = 256.

The second ingredient Escher needed was a nor-mal, undistorted drawing depicting the same scene:a gallery in which a print exhibition is held, one ofthe prints showing a seaport with quayside build-ings, and one of the buildings being the original

print gallery but reduced by a factor of 256. In orderto do justice to the varying amount of detail thathe needed, Escher actually made four studies in-stead of a single one (see [3]), one for each of thefour corners of the lithograph. Figure 5 shows thestudy for the lower right corner. Each of thesestudies shows a portion of the previous one (mod-ulo 4) but blown up by a factor of 4. Mathemati-cally we may as well view Eschers four studies asa single drawing that is invariant under scaling bya factor of 256. Square by square, Escher then fitted the straight square grid of his four studiesonto the curved grid, and in this way he obtainedPrentententoonstelling. This is illustrated inFigure 6.

Figure 2. A cyclic expansion expressed usingstraight lines.

Figure 3. A cyclic expansion expressed usingcurved lines.

Figure 4. Eschers grid.

Figure 5. One of Eschers studies.

Figure 6. Fitting the straight squares onto thecurved grid.

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Below, we shall imagine the undistorted pictureto be drawn on the complex plane C, with 0 in themiddle. We shall think of it as a functionf : C {black, white} that assigns to each z C itscolor f (z) . The invariance condition then expressesitself as f (256z) = f (z) , for all z C.

A Complex Multiplicative PeriodEschers procedure gives a very precise way ofgoing back and forth between the straight worldand the curved world. Let us make a number ofwalks on his curved grid and keep track of the cor-responding walks in the straight world. First, con-sider the path that follows the grid lines from Ato B to C to D and back to A . In the curved worldthis is a closed loop. The corresponding path,shown in Figure 7, in the straight world takes threeleft turns, each time travelling four times as far asthe previous time before making the next turn. Itis not a closed loop; rather, if the origin is prop-

erly chosen, the end point is 256 times the start-ing point. The same happens, with the same choiceof origin, whenever one transforms a single closedloop, counterclockwise around the center, fromthe curved world to the straight world. It reflectsthe invariance of the straight picture under a blow-up by a factor of 256.

No such phenomenon takes place if we do notwalk around the center. For example, start againat A and travel 5 units, heading up; turn left andtravel 5 units; and do this two more times. Thisgives rise to a closed loop in the curved world

depicted in Figure 8, and in the straight world itcorresponds to walking along the edges of a 5 5square, another closed loop. But now do the samething with 7 units instead of 5: in the straightworld we again get a closed loop, along the edgesof a 7 7 square, but in the curved world the pathdoes not end up at A but at a vertex A of the smallsquare in the middle. This is illustrated inFigure 9. Since A and A evidently correspond tothe same point in the straight world, any picturemade by Eschers procedure should, ideally, re-ceive the same color at A and A. We write ideally,since in Eschers actual lithograph A ends up inthe circular area in the middle.

We now identify the plane in which Escherscurved grid, or his lithograph, is drawn, also withC, the origin being placed in the middle. Define C by = A/A. A coarse measurement indi-cates that || is somewhat smaller than 20 and that the argument of is almost 3.

Replacing A in the procedure above by any pointP lying on one of the grid lines AB, BC, CD, DA,we find a corresponding point P lying on theboundary of the small square in the middle, andP will ideally receive the same color as P. Withinthe limits of accuracy, it appears that the quotient

New Escher MuseumIn November 2002 a new museum devoted to Escher's worksopened in The Hague, Netherlands. The museum, housed in thePalace Lange Voorhout, a royal palace built in 1764, contains anearly complete collection of Escher's wood engravings, etchings,mezzotints, and lithographs. The initial exhibition includes majorworks such as Day and Night, Ascending and Descending, andBelvedere, as well as the Metamorphoses and self-portraits. The mu-seum also includes a virtual reality tour that allows visitors to ridethrough the strange worlds created in Escher's works.

Further information may be found on the Web at http://www.escherinhetpaleis.nl/.

Allyn Jackson

Figure 7. The square ABCD transformed to thestraight world.

Figure 8. A 5 x 5 square transformed to thecurved world.

Figure 9. A 7 x 7 square transformed to thecurved world.

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P/P is independent of P and therefore also equalto . That is what we shall assume. Thus, when thesquare ABCD is rotated clockwise over an angleof about 160 and shrunk by a factor of almost 20,it will coincide with the small central square.

Let the function g, defined on an appropriatesubset of C and taking values in {black, white}, as-sign to each w its color g(w ) in Eschers lithograph.If Escher had used his entire gridwhich, towardsthe middle, he did notthen, as we just argued, onewould necessarily have g(P ) = g(P ) for all P asabove, and therefore we would be able to extendhis picture to all of C = C\{0} by requiringg(w ) = g(w ) for all w . This would not just fill inthe hole inside Eschers lithograph but also the im-mense area that finds itself outside its boundaries.

Elliptic CurvesWhile the straight picture is periodic with a multi-plicative period 256, an idealized version of the dis-torted picture is periodic with a complex period :

f (256z) = f (z), g(w ) = g(w ).

What is the connection between 256 and ? Canone determine other than by measuring it in Eschers grid?

We start by reformulating what we know. Forconvenience, we remove 0 from C and considerfunctions on C rather than on C. This leaves a holethat, unlike Eschers, is too small to notice. Next,instead of considering the function f with period256, we may as well consider the induced functionf : C/256 {black, white}, where 256 de-notes the subgroup of C generated by 256. Like-wise, instead of g we shall considerg: C/ {black, white}. Eschers grid providesthe dictionary for going back and forth between fand g. A moments reflection shows that all it doesis provide a bijection h: C/ C/256 suchthat g is deduced from f by means of composition:g = f h.

The key property of the map h is elucidated bythe quotation from Bruno Ernst, the originalsquares could better retain their square appear-ance: Escher wished the map h to be a conformalisomorphism, in other words, an isomorphism ofone-dimensional complex analytic varieties.

The structure of C/, for any C with|| = 1, is easy to understand. The exponentialmap C C induces a surjective conformal mapC C/ that identifies C with the universal cov-ering space of C/ and whose kernelL = Z2i + Z log may be identified with the fun-damental group of C/. Also, we recognizeC/ as a thinly disguised version of the ellipticcurve C/L.

With this information we investigate what themap h: C/ C/256 can be. Choosing co-ordinates properly, we may assume that h(1) = 1.

By algebraic topology, h lifts to a unique confor-mal isomorphism C C that maps 0 to 0 and in-duces an isomorphism C/L C/L256. A standardresult on complex tori (see [2, Ch. VI, Theorem4.1]) now implies that the map C C is a multi-plication by a certain scalar C that satisfiesL = L256 . Altogether we obtain an isomorphismbetween two short exact sequences:

0 L C exp C/ 0

h

0 L256 C exp C/256 0.

Figure 10 illustrates the right commutative square.In order to compute , we use that the multi-

plication-by- map L L256 may be thought ofas a map between fundamental groups; indeed, itis nothing but the isomorphism between the fun-damental groups of C/ and C/256 induced

exp exp

h

Figure 10. The picture on the lower left, drawn onC, is invariant under multiplication by i and by 4.Pulled back to C by the exponential map it gives riseto a picture that is invariant under translation by thelattice 14L256 = Zi/2+ Z log 4 . Pulling this pictureback by a scalar multiplication by = (2i + log 256)/(2i) changes the period latticeinto 14L = Zi/2+ Z(i log 4)/(i + 2 log 4) . Sincethe latter lattice contains 2i, the picture can nowbe pushed forward by the exponential map. Thisproduces the picture on the lower right, which isinvariant under multiplication by all fourthroots of .The bottom horizontal arrow represents themultivalued map w h(w ) = w = exp( logw ) ; it isonly modulo the scaling symmetry that it is well-defined.

450 NOTICES OF THE AMS VOLUME 50, NUMBER 4

by h. The element 2i in the fundamental groupL of C/ corresponds to a single counter-clockwise loop around the origin in C . Up to homotopy, it is the same as the path ABCDA alonggrid lines that we considered earlier. As we saw, Eschers procedure transforms it into a path in Cthat goes once around the origin and at the sametime multiplies by 256; in C/256 , this path be-comes a closed loop that represents the element2i + log 256 of L256 . Thus, our isomorphismL L256 maps 2i to 2i + log 256, and there-fore = (2i + log 256)/(2i) . The lattice L isnow given by L = 1L256, and from || > 1 wededuce

= exp(2i(log 256)/(2i + log 256)).= exp(3.1172277221 + 2.7510856371i).

The map h is given by the easy formulah(w ) = w = w (2i+log 256)/(2i) .

The grid obtained from our formula is given inFigure 11. It is strikingly similar to Eschers grid.Our small central square is smaller than Eschers;this reflects the fact that our value || .= 22.58 islarger than the one measured in Eschers grid. Thereader may notice some other differences, all ofwhich indicate that Escher did not perfectly achievehis stated purpose of drawing a conformal grid, butit is remarkable how close he got by his ownheadache-causing process.

Filling in the HoleIn order to fill in the hole in Prentententoonstelling,we first reconstructed Eschers studies from thegrid and the lithograph by reversing his own pro-cedure. For this purpose we used software speciallywritten by Joost Batenburg, a mathematics student

at Leiden. As one can see in Figure 12, the blankspot in the middle gave rise to an empty spiral inthe reconstructed studies, and there were other im-perfections as well. Next, the Dutch artists HansRichter and Jacqueline Hofstra completed and ad-justed the pictures obtained; see Figure 13.

When it came to adding the necessary grayscale,we ran into problems of discontinuous resolutionand changing line widths. We decided that the natural way of overcoming these problems was byrequiring the pixel density on our elliptic curve to

be uniform in the Haar measure. In practical terms,we pulled the sketches back by the exponentialfunction, obtaining a doubly periodic picture on C;it was in that picture that the grayscale was added,by Jacqueline Hofstra, which resulted in the dou-bly periodic picture in Figure 14. The completed ver-sion of Eschers lithograph, shown in Figure 15, wasthen easy to produce.

Other complex analytic maps h: C/ C/256 for various give rise to interestingvariants of Prentententoonstelling. To see these

Figure 11. The perfectly conformal grid.

Figure 12. Eschers lithograph rectified, bymeans of his own grid.

Figure 13. A detail of the drawing made byHans Richter and Jacqueline Hofstra.

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and to view animations zooming in to the centerof the pictures, the reader is encouraged to visit thewebsite escherdroste.math.leidenuniv.nl.

AcknowledgmentsThe assistance of Joost Batenburg, Cordon Art,Bruno Ernst, Richard Groenewegen, Jacqueline Hofstra, and Hans Richter is gratefully acknowl-edged. Cordon Art holds the copyright to all of M. C. Eschers works. The project was supportedby a Spinoza grant awarded by the Nederlandse Or-ganisatie voor Wetenschappelijk Onderzoek (NWO).

References[1] BRUNO ERNST, De toverspiegel van M. C. Escher, Meulen-

hoff, Amsterdam, 1976; English translation by John E.

Brigham: The Magic Mirror of M. C. Escher, BallantineBooks, New York, 1976.

[2] J. SILVERMAN, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.

[3] E. TH (design), The Magic of M. C. Escher, Harry N.Abrams, New York and London, 2000.

Figure 14. The straight drawing pulled back, by the complex exponential function, to a doublyperiodic picture, with grayscale added. The horizontal period is log 256, the vertical period is 2i.

Figure 15. The completed version of Eschers lithograph with magnifications of the center by factorsof 4 and 16.

452 NOTICES OF THE AMS VOLUME 50, NUMBER 4

Creating RepeatingHyperbolicPatternsOld andNewDouglas Dunham

For more than a hundred years, mathematicianshave drawn triangle tessellations in the Poincardisk model of the hyperbolic plane, and some artistshave found inspiration in these patterns. The Dutchgraphic artist M. C. Escher (18981972) was mostlikely the first one to be so inspired. He used clas-sical straightedge and compass constructions tocreate his hyperbolic patterns. More recently othershave used computer graphics to create hyperbolicpatterns. We will first explain how Escher drew the

underlying tessellations forhis patterns; then we willdescribe versions of thecomputer program thatgenerated the design for the2003 Mathematics Aware-ness Month poster.

Eschers MethodsIn 1958 H. S. M. Coxetersent Escher a copy of hispaper Crystal symmetryand its generalizations[Coxeter1]. In his replyEscher wrote, some ofthe text-illustrations andespecially Figure 7, page11, gave me quite a shock[Coxeter2].

Escher was shocked be-cause that figure showed him the long-desired so-lution to his problem of designing repeating pat-terns in which the motifs become ever smallertoward a circular limit. Coxeters Figure 7 con-tained the pattern of curvilinear triangles shown inFigure 1 (but without the scaffolding). Of course,that pattern can be interpreted as a triangle tes-sellation in the Poincar disk model of the hyper-bolic plane, although Escher was probably moreinterested in its pattern-making implications.

The points of the Poincar disk model are inte-rior points of a bounding circle in the Euclidean

plane. Hyperbolic lines are represented by diame-ters and circular arcs that are orthogonal to thebounding circle [Greenberg]. The Poincar diskmodel is attractive to artists because it is confor-mal (angles have their Euclidean measure) and itis displayed in a bounded region of the Euclideanplane so that it can be viewed in its entirety. Escherwas able to reconstruct the circular arcs in Coxetersfigure and then use them to create his first circlelimit pattern, Circle Limit I (Figure 2a), which he in-cluded with his letter to Coxeter. Figure 2b showsa rough computer rendition of the Circle Limit I de-sign. Eschers markings on the reprint that Coxetersent to him show that the artist had found thecenters of some of the circular arcs and drew linesthrough collinear centers [Schattschneider1].

The center of an orthogonal circular arc is ex-ternal to the disk and is called its pole. The locusof all poles of arcs through a point in the disk is aline called the polar of that point [Goodman-Strauss]. The external dots in Figure 1 are the polesof the larger arcs, and the external line segmentsconnecting them are parts of polars of the pointsof intersection of those arcs. Figure 1 shows oneinterior point and its polar as a large dot and a thickline on the left; a circular arc and its pole are sim-ilarly emphasized on the right.The external web ofpoles and polar segments is sometimes called thescaffolding for the tessellation. The fact that the po-lars are lines can be used to speed up the straight-edge and compass construction of triangle tessel-lations. For example, given two points in the disk,the center of the orthogonal arc through them isthe intersection of their polars.

Coxeter explained the basics of these techniquesin his return letter to Escher [Roosevelt], althoughby that time Escher had figured out most of this, as evidenced by Circle Limit I. Like Escher,mathematicians have traditionally drawn triangletessellations in the Poincar disk model usingstraightedge and compass techniques, occasionallyshowing the scaffolding. This technique was some-thing of a geometric folk art until the recentpaper by Chaim Goodman-Strauss [Goodman-Strauss], in which the construction methods werefinally written down.

For positive integers p and q , with 1/p+1/q < 1/2, there exist tessellations of the hyperbolicplane by right triangles with acute angles /p and/q . A regular p-sided polygon, or p-gon, can beformed from the 2p triangles about eachp-fold rotation point in the tessellation. Thesep-gons form the regular tessellation {p, q} byp-sided polygons, with q of them meeting at eachvertex. Figure 4 shows the tessellation {6,4} (with acentral group of fish on top of it). As can be seen,Escher essentially used the {6,4} tessellation inCircle Limit I.

Figure 1: The triangle tessellationof the Poincar disk that inspired

Escher, along with thescaffolding for creating it.

Douglas Dunham is professor of computer science at theUniversity of Minnesota, Duluth. His email address [email protected] Dunham designed the Mathemat-ics Awareness Month poster, which is available athttp://mathforum.org/mam/03/poster.html.

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Goodman-Strauss constructs the tessellation{p, q} in two steps. The first step is to constructthe central p-gon. To do this, he starts by con-structing a regular Euclidean p-sided polygon Pwith center O that forms the outer edges of the scaffolding. Then he constructs the hyperbolicright triangle with a vertex angle of /p at Oand its hypotenuse along a radius of P from O toone of the vertices A of P. The side of the righttriangle through O is part of a perpendicularbisector of an edge of P containing A . The bound-ing circle is easily determined from the right tri-angle. The vertices of the entire central p-gon canthen be constructed by successive (Euclidean) re-flections across the radii and perpendicular bisec-tors of edges of P. The second step is to constructall the other p-gons of the tessellation. This couldbe done by first inverting all the vertices in the cir-cular arcs that form the sides of the central p-gon,forming new p-gons, and then inverting vertices inthe sides of the new p-gons iteratively as manytimes as desired. But Goodman-Strauss describesa more efficient alternative method using factsabout the geometry of circles.

Our First MethodIn 1980 I decided to try to use computer graphicsto recreate the designs in each of Eschers fourCircle Limit prints (catalog numbers 429, 432, 434,and 436 in [Bool]). The main challenge seemed tobe finding a replication algorithm that would draweach copy of a motif exactly once. There were tworeasons for this. First, we were using pen-plottertechnology then, so multiple redrawings of thesame motif could tear through the paper. Effi-ciency was a second reason: the number of motifsincreases exponentially from the center, and inef-ficient algorithms might produce an exponentialnumber of duplications.

Moreover, I wanted the replication algorithm tobuild the pattern outward evenly in layers sothat there would be no jagged edges. At that timemy colleague Joe Gallian had some undergraduateresearch students who were working on findingHamiltonian paths in the Cayley graphs of finitegroups [Gallian]. I thought that their techniquescould also be applied to the infinite symmetrygroups of Eschers Circle Limit designs. This turnedout to be the case, although we found the desiredpaths in two steps.

The first step involved finding a Hamiltonianpath in the Cayley graph of the symmetry groupof the tessellation {p, q} . This was done by DavidWitte, one of Gallians research students. JohnLindgren, a University of Minnesota Duluth student,implemented the computer algorithm, with metranslating Wittes path into pseudo-FORTRAN[Dunham1].

The Cayley graphof a group G with aset of generators S isdefined as follows:the vertices are justthe elements of G ,and there is an edgefrom x to y if y = sxfor some s in S . Tech-nically, this defines adirected graph, but inour constructions theinverse of every ele-ment of s will also bein S , so for simplicitywe may assume thatour Cayley graphs areundirected. As an ex-ample, the symmetrygroup of the tessella-tion {p, q} is denoted[p, q]. That symmetrygroup is the same asthat of the tessella-tion by right triangleswith angles /p and/q . The standardset of generators forthe group [p, q] is{P,Q,R} , where P,Q , and R are reflec-tions across the tri-angle sides oppositethe angles /p, /q ,and /2 , respec-tively, in one such tri-angle.

There can be one-way or two-wayHamiltonian paths inthe Cayley graphs ofsymmetry groups ofhyperbolic patterns[Dunham3]. However,one-way paths aresufficient for our al-gorithms, so in thisarticle Hamiltonianpath will always de-note a one-way Ham-iltonian path.

There is a usefulvisual representationfor the Cayley graphsof the groups [p, q] and thus for their Hamiltonianpaths. A fundamental region for the tessellation{p, q} is a triangle that when acted on by the sym-metry group [p, q] has that tessellation as its orbit.This fundamental region can be taken to be a right

Figure 3: The Cayley graph of the group[6,4] with a Hamiltonian path.

Figure 2b: A computer rendition ofthe design in Eschers print CircleLimit I.

Figure 2a: The original Escher printCircle Limit I.

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triangle lying on the hori-zontal diameter to theright of center, with its/p vertex at the center ofthe disk. This triangle islabeled by the identity of[p, q]. Each triangle of thetessellation is then labeledby the group element thattransforms the funda-mental region to that tri-angle. Thus each trianglerepresents a group ele-ment. To represent anedge in the Cayley graph,we draw a line segmentconnecting the centers ofany two triangles sharinga side. Thus, there arethree line segments outof each triangle, each rep-resenting the reflectionacross one side.

Figure 3 shows aHamiltonian path in theCayley graph of the group[6,4] with the standardset of generators. Theheavy line segments, bothblack and gray, representthe Cayley graph; the lightlines show the triangle tes-sellation. The Hamilton-ian path consists of theheavy black line seg-ments. Essentially thesame path works for [p, q]in general, though slightdetours must be taken ifp = 3 or q = 3 [Dunham3].

Better MethodsA natural second stepwould have been to findHamiltonian paths in theCayley graphs of the sym-metry groups of EschersCircle Limit designs,which seemed possiblesince those symmetrygroups were all subgroupsof either [6,4] or [8,3].However, we took a

slightly bigger step instead, which also works forany symmetry group that is a subgroup of [p, q].This is the method used in the Mathematics Aware-ness Month poster. We first built up a supermotif(called the p-gon pattern in [Dunham1]) consistingof all the motifs adjacent to the center of the disk.

The supermotif for the Circle Limit I design isshown in Figure 4, superimposed on top of the{6,4} tessellation.

The second step then consisted of finding Hamil-tonian paths in Cayley coset graphs. Conceptu-ally, we form the cosets of a symmetry group bythe stabilizer H of its supermotif. Then, by anal-ogy with ordinary Cayley graphs, we define thevertices to be the cosets and say that there is anedge from xH to yH if yH = sxH for some s in S .

Again, there is a useful visual representationfor these coset graphs. The vertices correspond tothe p-gons in the tessellation {p, q} . The centralp-gon is labeled by H, and any other p-gon is la-beled by xH, where x is any element of the sym-metry group that maps the central p-gon to theother p-gon. In this case the generating set S is com-posed of words in the generators of the symmetrygroup. For each side of the central p-gon, there is atleast one word that maps the central p-gon acrossthat side. Much as before, we represent edges of thegraph as line segments between the centers of thep-gons. Figures 5 and 6 show the coset graph of asubgroup of [6,4]. Again, the graph edges are theheavy lines, either black or gray, and the light linesshow the {6,4} tessellation. The black graph edgesin Figure 5 show a Hamiltonian path.

One shortcoming of this method is that theHamiltonian path must be stored in computermemory. This is not a serious problem, since thepath can be encoded by small integers. That methodessentially works by forming ever-longer words inthe generators from the edges of the Hamiltonianpath. The transformations were represented byreal matrices, which led to roundoff error aftertoo many of them had been multiplied together toform the current transformation matrix. This wasa more serious problem.

Both problems were cured by using recursion.What was required was to find a Hamiltoniantreeor more accurately, a spanning treein thecoset graph. The tree is traversed on the fly by re-cursively transforming across certain sides of thecurrent p-gon. Those sides are determined by fairlysimple combinatorics [Dunham2] (there is an errorin the recursive algorithm in [Dunham1]). With thismethod the path from the root (central p-gon) tothe current p-gon is stored automatically in the re-cursion stack. Also, the recursion depth never getsdeeper than a few dozen, so there is no noticeableroundoff error from multiplying transformations.The black graph edges in Figure 6 show a spanningtree in the coset graph of [6,4].

Other researchers have used different methodsto generate a set of words in the generators of hy-perbolic symmetry groups in order to replicate re-peating hyperbolic designs. One method is to keeptrack of the transformations generated so far anditeratively add new transformations by multiplying

Figure 4: The central supermotif for Circle Limit I.

Figure 5: A Hamiltonian path in acoset graph of [6,4].

Figure 6: A spanning tree in a cosetgraph of [6,4].

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all the previous transformations by the group gen-erators, discarding duplicates. Then the desiredpattern is generated by applying the final set oftransformations to the motif. The techniques of au-tomatic groups have been used to efficiently gen-erate nonredundant sets of words. Silvio Levy hasused this technique to create a computer renditionof Eschers Circle Limit III [Levy].

Like many mathematicians, I was immediatelyenthralled by M. C. Eschers intriguing designswhen I first saw them more than thirty years ago.Several years later I started working in the field ofcomputer graphics, and that medium seemed likean obvious one to use to produce Escher-like de-signs. When discussing the symmetry groups ofEschers hyperbolic patterns with Joe Gallian, itoccurred to me that Hamiltonian paths could beused as a basis for an algorithm to generate suchpatterns. At this point everything had come to-gether, and I could not resist the temptation to re-generate Eschers hyperbolic patterns with a graph-ics program. As described above, the students andI achieved this goal.

Having gone to the trouble of implementing ahyperbolic pattern program, I could not resist thefurther temptation of creating more hyperbolicpatterns. I found inspiration in Eschers Euclideanrepeating patterns. In constructing my patterns, Inoticed I was paying attention to aesthetic issuessuch as shape and color. More of these hyperbolicdesigns can be seen in my chapter in the recentbook Eschers Legacy [Schattschneider2]. When designing these patterns, I think of myself as working in the intersection of interesting mathe-matics, clever algorithms, and pleasing art.

AcknowledgementsI would like to thank Doris Schattschneider for herconsiderable help, especially with the history ofEscher and Coxeters correspondence. I would alsolike to thank the many students who have workedon the programs over the years. Finally, I would liketo thank Abhijit Parsekar for help with the figures.

References[Bool] F. H. BOOL, J. R. KIST, J .L. LOCHER, and F. WIERDA, ed-

itors, M. C. Escher, His Life and Complete GraphicWork, Harry N. Abrahms, Inc., New York, 1982.

[Coxeter1] H. S. M. COXETER, Crystal symmetry and itsgeneralizations, Royal Soc. Canada (3) 51 (1957), 113.

[Coxeter2] , The non-Euclidean symmetry of Eschers picture Circle Limit III, Leonardo 12 (1979),1925, 32.

[Dunham1] D. DUNHAM, J. LINDGREN, and D. WITTE, Creatingrepeating hyperbolic patterns, Comput. Graphics 15(1981), 7985.

[Dunham2] D. DUNHAM, Hyperbolic symmetry, Comput.Math. Appl. Part B 12 (1986), no. 1-2, 139153.

[Dunham3] D. DUNHAM, D. JUNGREIS, and D. WITTE, InfiniteHamiltonian paths in Cayley digraphs of hyperbolicsymmetry groups, Discrete Math. 143 (1995), 130.

[Gallian] J. GALLIAN, Online bibliography of the DuluthSummer Research Programs supervised by JosephGallian, http://www.d.umn.edu/~jgallian/progbib.html.

[Goodman-Strauss] CHAIM GOODMAN-STRAUSS, Compass andstraightedge in the Poincar disk, Amer. Math. Monthly108 (2001), 3849.

[Greenberg] MARVIN GREENBERG, Euclidean and Non-EuclideanGeometries, 3rd Edition, W. H. Freeman and Co., 1993.

[Levy] SILVIO LEVY, Escher Fish, at http://geom.math.uiuc.edu/graphics/pix/Special_Topics/Hyperbolic_Geometry/escher.html.

[Roosevelt] The letter of December 29, 1958, from H. S. M.Coxeter to M. C. Escher, from the Roosevelt collectionof Eschers works at The National Gallery of Art, Wash-ington, DC.

[Schattschneider1] DORIS SCHATTSCHNEIDER, A picture of M. C.Eschers marked reprint of Coxeters paper [1], privatecommunication.

[Schattschneider2] DORIS SCHATTSCHNEIDER and MICHELEEMMER, editors, M. C. Eschers Legacy: A CentennialCelebration, Springer-Verlag, 2003.

Review of M. C.Eschers Legacy: A CentennialCelebrationReviewed by Reza Sarhangi

M. C. Eschers Legacy: A Centennial CelebrationMichele Emmer and Doris Schattschneider, EditorsSpringer, 2003450 pages, $99.00ISBN 3-540-42458-X

A few years ago an old friend who had visited aremote village in South America brought me a blan-ket as a souvenir. The purpose of this souvenir wasto introduce me to a tessellation design of a for-eign culture. To my amusement, the design was awork of Escher! The Dutch artist Maurits CornelisEscher (18981972), who was inspired by the ar-chitecture of southern Italy and the pattern designsof North African Moors, is the source of inspira-tion and fascination of an incredible number ofknown and unknown artists in various culturesaround the world.

Reza Sarhangi is the director of the International Con-ference of Bridges: Mathematical Connections in Art, Music,and Science ( http://www.sckans.edu/~bridges/). Heis also the graduate program director for mathematics education at Towson University. His email address [email protected]

http://www.d.umn.edu/~jgallian/progbib.htmlhttp://www.d.umn.edu/~jgallian/progbib.htmlhttp://geom.math.uiuc.edu/graphics/pix/Special_Topics/Hyperbolic_Geometry/escher.htmlhttp://geom.math.uiuc.edu/graphics/pix/Special_Topics/Hyperbolic_Geometry/escher.htmlhttp://geom.math.uiuc.edu/graphics/pix/Special_Topics/Hyperbolic_Geometry/escher.htmlhttp://www.sckans.edu/~bridges/

456 NOTICES OF THE AMS VOLUME 50, NUMBER 4

In the introduction of the book M. C. Escher, TheGraphic Work, Escher describes himself: The ideasthat are basic to [my works] often bear witness tomy amazement and wonder at the laws of naturewhich operate in the world around us. He whowonders discovers that this is in itself a wonder.By keenly confronting the enigmas that surroundus, and by considering and analyzing the obser-vations that I had made, I ended up in the domainof mathematics. Although I am absolutely withouttraining or knowledge in the exactsciences, I often seem to have morein common with mathematicians thanwith my fellow artists. This de-scription may very well explain whyan international congress held in Juneof 1998 in Rome and Ravello, Italy, tocelebrate the centennial of the birthof Escher was organized by two math-ematicians, Michele Emmer and DorisSchattschneider. This conference re-sulted in an immensely interestingcollection of articles presented in thebook M. C. Eschers Legacy, A Cen-tennial Celebration.

Emmer and Schattschneider, whoalso edited the book, are well knownto mathematicians, scientists, andartists who seek aesthetic connec-tions among disciplines. Schattschneider, a math-ematics professor at Moravian College, has writtennumerous articles about tessellations and Eschersworks as well as the book Visions of Symmetry(W. H. Freeman and Co., 1992), which describesEschers struggles with the problem of dividing theplane, and his achievements. Emmer, a mathemat-ics professor at the University of Rome, is one ofthe first in our time to call for a gathering of math-ematicians and artists under one roof. He edited thebook The Visual Mind (MIT Press, 1993) and createdin the 1970s a series of videos about connectionsbetween mathematics and art. Several prominentscientists, such as geometer H. S. M. Coxeter andphysicist and mathematician Roger Penrose, ap-peared in those videos. One of the videos, TheFantastic World of M. C. Escher, is still currently avail-able.

This book is divided into three parts. The firstpart, Eschers World, includes articles by authorswith vast records of intellectual activities in con-nections among disciplines and in Eschers works.To mention a few, I should name Bruno Ernst, whowrote The Magic Mirror of M. C. Escher in 1976, andDouglas Hofstadter, who wrote Gdel, Escher, Bach:An Eternal Golden Braid in 1979. The book comeswith a CD-ROM that complements many of theforty articles. The CD-ROM contains illustrationsof artwork by contemporary artists who made con-tributions to the book, as well as several videos, a

video-essay based on an Escher letter, some ani-mations, and a puzzle.

Some articles in the first part of the book demon-strate dimensions of Escher that are mainly un-known to the public and therefore have not beenextensively investigated. Eschers Sense of Won-der by Anne Hughes and In Search of M. C. Es-chers Metaphysical Unconscious by Claude Lam-ontagne are two notable examples.

We also read about an Escher admirer and avidcollector of his works, Cornelius Roo-sevelt, a grandson of American pres-ident Theodore Roosevelt. The arti-cle, by J. Taylor Hollist and DorisSchattschneider, illustrates a deeprelationship between the artist andthe art collector and describes howRoosevelt on many occasions playedan advisory role for Escher in regardto the exhibitions and publications ofthe artists works in the U.S. Cor-nelius Roosevelt donated his collec-tion to the National Gallery of Art inWashington, DC. The collection is ac-cessible to researchers, educators,and other interested individuals.

The second part, Eschers ArtisticLegacy, includes works of artists and

authors such as S. Jan Abas, Victor Acevedo, RobertFathauer, and Eva Knoll. There are also contribu-tions by the artist-mathematician Helaman Fergu-son and by Marjorie Rice, a homemaker withoutmathematical background who was inspired by theidea of pentagonal tessellations of the plane andwho discovered four tilings unknown to the mathe-matics community at the time.

Not all the artists contributing to this part of thebook were inspired by Escher; it is not a collectionof papers written by the artists disciples. Rather,the papers illustrate how the same sources forEschers inspiration inspired these artists in a par-allel way. Ferguson expressed that both he andEscher responded aesthetically to the same source:mathematics. Abas in a similar manner pointed toIslamic patterns as a common source. The readernot only becomes more familiar with the sourcesof inspiration but also has the chance to observehow the same sources resulted in different ap-proaches and expressions in the final products ofdifferent artists. This helps the reader to reach adeeper understanding of Eschers works.

The third part, Eschers Scientific and Educa-tional Legacy, presents articles by the computer scientist Douglas Dunham, who is famous for per-forming tilings of the hyperbolic disk; the geometerH. S. M. Coxeter; the author and editor Istvn Hargittai;Kevin Lee, who created the software utility Tessel-lation Exploration; and more.

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Even though the majority of the articles in thebook are accessible to a reader with little back-ground in mathematics, there are works in partthree that require familiarity with such conceptsas geodesic, hyperbolic geometry; symmetrygroups; and densest packing. These articles may beconsidered as a resource for research in under-graduate or graduate mathematics and mathe-matics education.

Escher is especially well known for two types ofworks: impossible structures (an idea borrowedfrom Penrose) and the regular division of the plane(influenced by Moorish artists of the Alhambra inGranada, Spain). Through impossible structuresEscher challenges our concepts of the real worldby tweaking our perception of dimensions. Andthrough Eschers other trademark, the regulardivision of space and tessellations, he expressesideas such as duality, symmetry, transformations,metamorphosis, and underlying relations amongseemingly unrelated objects. The book presentsseveral articles that address these two categories.However, some authors go further and study ex-amples of Eschers works that go beyond thesecategories.

Bruno Ernst correctly suggests that the attrac-tion of people to impossible structures and regu-lar divisions of the plane has given a one-sided andincorrect image of Escher that ignores what hewanted to convey through his prints. Ernst notesthat Escher was not a depicter of optical illusions,which are very different from the impossible con-structions; this is a misconception on the part ofthe public. An example Ernst presents shows howthe intention of the artist may be very differentfrom the perception of the spectator.

Hofstadter writes about the first time he saw anEscher print. He was twenty years old in January1966, and the print was in the office of Otto Frisch,who played a major role in unraveling the secretsof nuclear fission. Hofstadter was mesmerized byEschers work Day and Night, showing two flocksof birds, one in white and one in black, flying inopposite directions. He asked Frisch, What is this?and Frisch replied, It is a woodcut by a Dutchartist, and I call it Field Theory. . . .

The young man ponders the relationship tophysics that this artwork may have sparked inFrischs mind: I knew that one of the key princi-ples at the heart of field theory is the so-called CPTtheorem, which says that the laws of relativisticquantum mechanics are invariant when three flipsare all made in concert: space is reflected in a mir-ror, time is reversed, and all particles are inter-changed with their antiparticles. This beautifuland profound principle of physics seemed deeplyin resonance with Frischs Escher print . . . .

The above example and more throughout thebook illustrate the role of the artist Escher: a precise

observer and a tireless and deep thinker whose printsnot only bring us a sense of fascination and admi-ration but also provoke our intelligence to discovermore relations beyond the scope and the knowledgeof the artist through the meticulous and detailed pre-sentations of symmetry, duality, paradox, harmony,and proportion. It is, then, not unrelated that AlbertFalcon, a professor at the cole des Beaux Arts in Paris,in a 1965 paper classified Escher among the think-ing artists, along with Da Vinci, Drer, and Piero dellaFrancesca.

I would like to end this review with the samepoem that Emmer included at the end of one of hisarticles in this book. It is a Rubaiyat stanza by thePersian mathematician and poet of the eleventh century, Omar Khayym.

Ah, moon of my delight, that knows no waneThe moon of Heaven is rising again,How oft hereafter rising shall she lookThrough this same garden after us in vain!

Although Escher himself is no longer among us,M. C. Eschers Legacy, like a garden of continuallyblooming flowers, allows us to appreciate his her-itage anew.

About the CoverThis months cover exhibits the recently

constructed extension, described in the articleby Lenstra and de Smit, of M. C. Eschers ex-traordinary lithograph Print Gallery.

Bill Casselman([email protected])