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be replaced by “tile” or“shape”. Haag’s definition (with Escher’samendments) wasadopted by Escher andwould guide all of hissymmetry investigations. He later carefullyrecorded the definition on the back of hissymmetry drawing 25(1939) of lizards (thedrawing is depictedin Escher’s lithographReptiles).Pólya’s article [43]would have a greatinfluence on Escher.Escher carefully copied (by hand, in ink) theFigure 1. A copy of the display in [43], full text that outlinedsigned by Pólya. the four isometriesof the plane and announced Pólya’s classification of periodic planartilings by their symmetry groups. Pólya was evidently unaware that this classification had beencarried out by Fedorov more than thirty yearsearlier. Escher was already intuitively aware ofthe congruence-preserving transformations Pólyaspoke of but probably didn’t understand any of thediscussion about symmetry groups. What struckhim was Pólya’s full-page chart that displayedan illustrative tiling for each of the seventeenplane symmetry groups (Figure 1). Escher carefully sketched each of these seventeen tilings in acopybook and studied them, map-coloring someof them [47]. Among these, there were tilings thatdisplayed symmetries he had not recorded in theAlhambra; for example, tilings whose only symmetries other than translations were glide-reflectionsor fourfold (90 ) and twofold (180 ) rotations. Withinone month of studying these, Escher had completedhis first symmetry drawings displaying fourfold rotation symmetry: squirming lizards interlocked four ata time, pinwheeling where four feet met [53, p. 127].He featured a portion of one of these drawings at thecenter of his woodcut Development I, completed inthe same month.Escher was so grateful for the help that Pólya’spaper provided that he wrote to the mathematicianto thank him. He sent Pólya the print Development Iand asked the mathematician whether or not he hadwritten a book on symmetry for “laymen” as his articleindicated he had hoped to do. Although a writer oncecharacterized Pólya’s reply as polite but formal,indicating he hadn’t written the hoped-for book[53, p. 22], Pólya wrote to me in 1977 that he andEscher had corresponded more than once andthat he regretted losing the correspondence inhis haste to come to America in 1940. A recent708NOTICESOF THEdiscovery of a forgotten suitcase full of Pólya’snotes and other collected letters and papers, nowin the Pólya archives at Stanford University, showsthat Pólya even sent Escher his own attempt at anEscher-like tiling. Among these papers is Pólya’sdrawing of a tiling by snakes, inscribed “sent toMCE”, at the address where Escher resided from1937 to 1940. Also, there is an outline of Pólya’snever-completed book The Symmetry of Ornamentand many sketches of tilings, both for the plannedbook and for the 1924 article that so influencedEscher [53, pp. 335–36].Escher as a Mathematical ResearcherFrom 1937 to 1941 Escher plunged into a methodical investigation that can only be termed mathematical research. Haag’s article had given him adefinition of “regular division of the plane”, andPólya’s article showed him that there were manytile shapes that could produce these. He wanted tofind more and characterize them. The questions hepursued, using his own techniques, were:(1) What are the possible shapes for a tile thatcan produce a regular division of the plane, thatis, a tile that can fill the plane with its congruentimages such that every tile is surrounded in thesame manner?(2) Moreover, in what ways are the edges of sucha tile related to each other by isometries?The only isometries that Escher allowed in orderto map a tile to an adjacent tile were translations,rotations, and glide-reflections—a reflection wouldrequire a tile’s edge to be a straight segment,not a natural condition for his creature tiles. In1941–1942 he recorded his many findings in adefinitive Notebook that was to be his private encyclopedia about regular divisions of the plane andhow to produce and color them [46], [48], [53]. TheNotebook had two parts: its cover inscribed (heretranslated) “Regular divisions of the plane intoasymmetric congruent polygons; I Quadrilateralsystems MCE 1-1941 Ukkel; II Triangle systemsX-1942, Baarn”.Escher’s study of “quadrilateral systems” wasextensive. He represented these tilings symbolically with a grid of congruent parallelograms inwhich each parallelogram represented a single tile.He shaded the grids checkerboard style, so thateach parallelogram shared edges only with parallelograms of the opposite color. He was interestedin asymmetric tiles (after all, his creature tileswere to be primarily asymmetric), and in order toindicate the asymmetry, placed a hook inside eachparallelogram. The hook provided orientation,while small circles and squares on the tile’s boundary indicated twofold and fourfold centers aboutwhich the tile could rotate into an adjacent tile.Escher was aware that certain symmetries requiredspecial parallelogram grids and so consideredfive different categories: arbitrary parallelogram,AMSVOLUME 57, NUMBER 6

All M. C. Escher works 2009 The M. C. Escher Company–The Netherlands.All rights reserved. Used by permission. www.mcescher.com.Figure 2. A copybook page showing Escher’s method of investigation of regular divisions of theplane. His symbolic notation is explained in our text.rhombus, rectangle, square, and isosceles righttriangle (a grid of squares in which the diagonalshave been drawn). He labeled these A–E, respectively. As he sought to answer his two questions,he filled the pages of several school copybookswith his sketches of marked grids representingtilings, scratching out those that didn’t work outor that duplicated an earlier discovery. Each timehe discovered a marked grid that represented aregular division of the plane, he recorded it andmade an example of a tiling with a “shaped tile”,its vertices marked by letters.To quickly record how each edge of a tile wasrelated to another edge of the same tile or anadjacent tile, Escher devised his own notation: meant “related by a translation” and “related bya glide-reflection”. An S on its side meant “relatedby a 180 rotation” and L meant “related by a 90 rotation”. Figure 2 shows one copybook page withfive different “rhombus systems” on the left andshaped tilings for two of these systems on theright. Note Escher’s “voorbeeld maken!” at the bottom of the page—“make an example!” His resultswere recorded entirely visually, with no need forwords. Ultimately he found ten different classesof these tilings and numbered the classes I – X. HisNotebook charts giving both visual and descriptiveversions of the classes are in [53, pp. 58–61].To discover still other regular divisions, thosefor which three colors would be required for mapcoloring, Escher employed a technique that hecalled “transition”. Figure 3 recreates one of hisJUNE/JULY 2010examples. He would begin with a two-color regulardivision from one of his ten categories (Figure 3begins with type IIA). Each of these categories hadfour tiles meeting at every vertex and requiredonly two colors. He would then choose a tile anda segment of the boundary that connected oneof its vertices (say B) to another carefully chosenboundary point (say A) that was not a vertex of thetiling (Figure 3a). Using A as a pivot point, he wouldthen pivot the boundary segment connecting Aand B (stretching it if necessary) so that vertex Bslid along the boundary of the tile, stopping at anew position (say C). Repeating this on the corresponding segments of the boundaries of all tilesproduced a new tiling with vertices at which threetiles met, requiring three colors for map-coloring(Figure 3b). The process could be continued withthe new segment AC, sliding C along the boundaryuntil it reached a vertex D of the original tiling.This produced a new tiling that again required onlytwo colors (Figure 3c). At the intermediate (3-color)stage, the network of tile edges was certainly nothomeomorphic to the original, but surprisingly,at the end (2-color) stage, the new network of tileedges might also not be homeomorphic to that ofthe original tiling. Escher thought of the intermediate (3-color) tiling as having components of boththe beginning and ending 2-color tilings, and solabeled it with both types. In Figure 3, his typeIIA system is transformed to IIA–IIIA, and that istransformed to system IIIA. In this instance, thetiles in the final tiling have three, not four, edgesNOTICESOF THEAMS709

and meet six at a vertex;Escher noted that this wasan exceptional case [53,p. 62].Escher did not recordthese discoveries withwords, but in his Notebook he displayed sixteenpages of carefully drawnillustrations of transitions that cover all of histen categories [53, pp.62–69]. In many cases,he discovered more thanone distinct transition ofthe same tiling. Using today’s terminology, he discovered how to producetilings of different isohedral types beginning witha single isohedral tiling.And he also recorded ina chart (a digraph!) whichof his ten categories led toothers. This chart makesclear that his process oftransition can change thetopological and combinatorial properties of atiling but not change itssymmetry group [48], [53,p. 60].Figure 3. Escher’s transitionThe last section ofprocess takes (a) a 2-colored tiling Escher’s “quadrilateralwith 180 rotation centers at tile systems” study summavertices and midpoints of edges, rizes in ten pages his into (b) a 3-colored tiling, to (c) a vestigations of what he2-colored tiling with 180 rotation called “2-motif” tilingscenters at midpoints of tile edges. [53, pp. 70–76]. He wouldbegin with a regular division, map-colored with two or three colors, thensplit each tile (in the same manner) into two distinct shapes, so that the resulting tiling could becolored with two colors. This investigation wasspurred by his fascination with what he called“duality”— many of his prints play with the idea ofinterchanging the role of figure and ground, or juxtaposition of opposites. Sky and Water I and CircleLimit IV (Angels and Devils) are famous examples.For example, in Sky and Water I (Figure 4) ahorizontal row of interlocked flat tiles at the centeralternates white fish and black birds, dividing theprint into upper and lower halves (sky and water).The fish in this center row can serve as figure andthe birds as ground, or vice versa. But as the eyemoves upward from this row of tiles, the creaturesseparate and take on distinct roles. The black birdsbecome three-dimensional as they rise, while thewhite fish melt to become sky. The fish becomethe background against which figures of birds fly.710NOTICESOF THEAs the eye moves downward from the center rowof tiles, the opposite transformation takes place.Now the fish gain three-dimensional form and theblack birds dissolve to become water in which thefish swim. In mathematics, the essence of dualobjects is that each completely defines the other,such as a set and its complement, a statement andits negation. In addition to the figure/ground duality, there are other kinds of duality represented inthis single print: black and white, sky and water.And opposites: bird and fish often denote opposites (think “neither fish nor fowl”), and in theprint, each bird is placed exactly opposite a fish,with the invisible surface of the water acting as acompositional mirror.Part II of Escher’s Notebook is brief, devoted towhat he called “triangle systems”—regular divisionshaving 120 rotation centers (system A) or 60 , 120 ,and 180 rotation centers (system B). After explaining the necessary placement of rotation centers, herecords only twenty different tilings, several withtwo motifs, and all carefully map-colored to respectsymmetry. Most require three colors. Unlike in hisquadrilateral systems, some of his tiles have rotation symmetry, and from these he derives other tileswith one or two motifs [53, pp. 79–81].In 1941, as he was nearing the end of theseinvestigations, Escher and his family moved toBaarn, Holland, where he would spend all but thelast two years of his life. In the years following, heproduced more than 100 regular divisions of theplane, each final version numbered and carefullydrawn on graph paper, its creature tiles outlinedin ink, map-colored using watercolors, respectingthe symmetries of the tiling. As his portfolio ofsymmetry drawings grew, he referred to it as his“storehouse”. Fragments of these drawings wouldbe featured in many prints, notecards, exhibit announcements, painted and tiled public works, andeven carved on the surface of a ball. In all, thereare 134 numbered symmetry drawings and manyunnumbered sketches.Escher carried out several other minor mathematical investigations in order to achieve certaineffects in his art. Some of these results wererecorded in a notebook entitled Regular Divisionof the Plane: Abstract Motifs, Geometric Problems,and others were gathered in small folios. He studied several Moorish-like tilings and investigatedlinked rings (seen in his last print, Snakes). Heenumerated several tilings by congruent triangleswhile designing bank notes. He also recordedtwo theorems he evidently discovered but didnot prove. One was about concurrent lines in atriangle, and the other about concurrent diagonals in a special tiling hexagon [53, pp. 82–93].At my request, the first theorem was verified byA. Liu and M. Klamkin [35] and the second byJ. F. Rigby [44].AMSVOLUME 57, NUMBER 6

All M. C. Escher works 2009 The M. C. Escher Company–The Netherlands.All rights reserved. Used by permission. www.mcescher.com.An investigation in 1942 that was an amusement, shared with his children and grandchildren[18, pp. 9–11], [50], was combinatorial—to determine how many different patterns could be generated by following this algorithm:Decorate a square with an asymmetric motif and use four copies of thedecorated square (independently chosenfrom any of four rotated aspects) to fillout a 2 2 larger square, then translatethe larger square in the direction of itsedges to fill the plane.With a methodical search of the 44 possible 2 2filled squares, eliminating obvious duplications,and by sketching examples with a simple motif forthe rest, he ultimately found twenty-three distinctpatterns. That is, no two of these twenty-three patterns were identical, allowing rotations. He alsoasked the combinatorial question in two specialcases in which reflected aspects of the decoratedsquare were also allowed. In these cases, thechoices of the four copies of the decorated squarewere restricted as follows:Case (1)—two choices must be the samerotated aspect and independently, theother two choices the same reflectedaspect. Case (2)—two choices must bedifferent rotated aspects and independently, the other two choices differentreflected aspects.Escher’s results were sketched in copybooks andlater printed with inked carved wooden stampsusing a motif that produced patterns resemblingknitted or crocheted pieces. He also made a “ribbon” design, outlining crossing bands in a square,and carved four wooden stamps—one of the original design, one of its reflection, and two others thatreversed the crossings in the first two stamps. Hedid not attempt to find the number of patternsproduced by the 416 possible 2 2 squares filledwith aspects of these, but he did produce severalpatterns with them and colored them with a minimum number of colors so that continuous ribbonstrands had the same color and no two bands ofthe same color ever crossed [53, pp. 44–52], [17,p. 41].Escher’s Interactions with MathematiciansUntil 1954 few mathematicians outside of Hollandknew of Escher’s work. That year the InternationalCongress of Mathematicians (ICM) was held inAmsterdam, and N. G. de Bruijn arranged for anexhibit of Escher’s prints, symmetry drawings,and carved balls at the Stedelijk Museum [11]. Hewrote in the catalog, “Probably mathematicians willnot only be interested in the geometrical motifs;the same playfulness which constantly appearsin mathematics in general and which, to a greatJUNE/JULY 2010Figure 4. M. C. Escher’s Sky and Water I, 1938. Woodcut,435 cm x 439 cm.many mathematicians is thepeculiar charm of their subject, will be a more importantelement” [2].When Roger Penrose visitedthe exhibit, he was amazedand intrigued. Escher’s printRelativity especially caughthis eye. It shows three prominent staircases in a triangular arrangement (and somesmaller staircases), as seenfrom many different viewpoints, with several personssimultaneously climbing or Figure 5. Penrose’s tribar.descending them in an impossible manner, defying the law of gravity. Penrosewas inspired to find a structure whose parts wereindividually consistent but, when joined, became“impossible”. After returning to England he cameup with the idea of the now-famous Penrose tribarin which three mutually perpendicular bars appearto join to form a triangle (Figure 5). Following that,his father devised an “endless staircase”, anotherobject that can be drawn on paper but is impossible to construct as it appears [41, pp. 149–50].Penrose then closed the loop of discovery bysending the sketches of these impossible objectsto Escher, who in turn used them in crafting theperpetual motion in his print Waterfall and thenever-ending march of the monks in Ascendingand Descending.Penrose also visited Escher’s home in 1962 andbrought a gift of identical wooden puzzle piecesNOTICESOF THEAMS711

derived from a 60 rhombus.Escher soon sent Penrose thepuzzle’s solution, enclosinga sketch of the unique wayin which the pieces fitted together. Here, congruent tileswere surrounded in two distinct ways. In 1971 Escherproduced his only tiling withone tile that was not a regulardivision (today it would becalled 2-isohedral). It was thelast of his numbered symmetry drawings, with a littleghost that filled the planeaccording to the rules of Penrose’s puzzle [41, pp. 144–45;Figure 6. Coxeter’s53, p. 229].Figure 7, with Escher’sH. S. M. Coxeter also sawmarkings (here computerEscher’s work for the firstenhanced for visibility).time during that ICM in 1954,and upon returning to Canada, he wrote Escher a letter to express his appreciation of the artist’s work. Three years later, hewrote again to ask if he might use two of Escher’ssymmetry drawings to illustrate an article basedon his presidential address to the Royal Societyof Canada. The article discussed symmetry in theEuclidean plane and also in the Poincaré disk modelof the hyperbolic plane and on a sphere surface [3].Escher readily agreed, and when he later received areprint of the article, he wrote to Coxeter, “some ofthe text-illustrations and especially figure 7, page11, gave me quite a shock” [5, p. 19]. The figure’shyperbolic tiling, with triangular tiles diminishing insize and repeating (theoretically) infinitely withinthe confines of a circle, was exactly what Escherhad been looking for in order to capture infinityin a finite space.Escher worked over the figure with compassand straightedge and circled important points(Figure 6). From this, he managed to discernenough of the geometry to produce his print CircleLimit I. But he wanted to know more, and sent alarge diagram to Coxeter showing what he hadfigured out, namely, the location of centers of sixof the circles (Figure 7). In his letter, he politelyasked Coxeter for “a simple explanation how toconstruct the [remaining] circles whose centresapproach [the bounding circle] from the outsidetill they reach the limit.” He also asked, “Are thereother systems besides this one to reach a circlelimit?” [5, p. 19], [54, p. 263]. Coxeter replied witha minimal answer to Escher’s first request:The point that I have marked on yourdrawing (with a red o on the back ofthe page) lies on three of your circleswith centres 1, 4, 5. These centrestherefore lie on a straight line (whichI have drawn faintly in red) and thefourth circle through the red pointmust have its centre on this same redline. [54, p. 264]From this, Escher was supposed to constructthe complete scheme. By contrast, Coxeter answered the second question at length, beginning,“Yes, infinitely many! This particular pattern isdenoted by [4, 6]” and then explained for whichp and q patterns [p, q] exist, referring to the textFigure 7. Escher’s diagram sent to Coxeter, exhibiting what the artist had figured out. The originaldrawing is faint, drawn in pencil on tracing paper. This is a reconstruction by the author, andshows Coxeter’s red markings.712NOTICESOF THEAMSVOLUME 57, NUMBER 6

Generators and Relations, and enclosing a “spare3copy of 7 ” [54, p. 264].Escher was disappointed with this reply, yet itonly increased his determination to figure thingsout. He wrote to his son George:[Coxeter] encloses an example of usingthe values three and seven, of all things!However this odd seven is of no use tome at all; I long for two and four (orfour and eight) My great enthusiasmfor this sort of picture and my tenacityin pursuing the study will perhaps leadto a satisfactory solution in the end. it seems to be very difficult for Coxeterto write intelligibl

in Rome, where two sons were born. Until 1935 he continued to make frequent sketching trips, most in southern Italy, returning to his Rome studio to compose his sketches for woodcuts and litho-graphs. In 1935, with the growing rise of Fascism in Italy and his sons in ill health, Escher felt it best to move his family from Italy to Switzerland. In