capturing infinity

8/10/2019 Capturing Infinity

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Capturing Innity

march 2010


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In July 1960, shortly after his 62nd birthday, the graphic artist M.C.Escher completed Angels and Devils,the fourth (and nal) woodcut in hisCircle Limit Series. I have a vivid memory of my rst view of a print ofthis astonishing work. Following sensations of surprise and delight, twoquestions rose in my mind. What is the underlying design? What is thepurpose? Of course, within the world of Art, narrowly interpreted, onemight regard such questions as irrelevant, even impertinent. However,for this particular work of Escher, it seemed to me that such questionswere precisely what the artist intended to excite in my mind.

In this essay, I will present answers to the foregoing questions, basedupon Eschers articles and letters and upon his workshop drawings. For themathematical aspects of the essay, I will require nothing more but certainlynothing less than thoughtful applications of straightedge and compass.

Capturing InnityTe Circle Limit Series of M.C. Escher BY THOMAS WIETING

Escher completedCLIV , alsoknown asAngels and Devils, in 1960.

The Dutch aMaurits C. E

(18981

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Capturing InnityIn 1959, Escher described, in retrospect, atransformation of attitude that had occurredat the midpoint of his career:

I discovered that technical mastery was nolonger my sole aim, for I was seized by anotherdesire, the existence of which I had never sus-pected. Ideas took hold of me quite unrelated tographic art, notions which so fascinated me thatI felt driven to communicate them to others.Te woodcut called Day and Night, com-

pleted in February 1938, may serve as a sym-

bol of the transformation. By any measure, itis the most popular of Eschers works.

Prior to the transformation, Escher pro-duced for the most part portraits, landscapes,and architectural images, together with com-mercial designs for items such as postagestamps and wrapping paper, executed at anever-ris ing level of technical skill. However,following the transformation, Escher producedan inspired stream of the utterly original worksthat are now iden tied with his name: the illu-sions, the impossible gures, and, especially,the regular divisions (called tessellations) ofthe Euclidean plane into potentially innitepopulations of sh, reptiles, or birds, of statelyhorsemen or dancing clowns.

Of the tessellations, he wrote:This is the richest source of inspiration that Ihave ever struck; nor has it yet dried up.However, while immensely pleased in prin-

ciple, Escher was dissatised in prac tice witha particular feature of the tessellations. Hefound that the logic of the underlying patterns

Day and Night (1938) is the most popular of Eschers works.

Figure A

Regular Division III (1957) demonstrates Escmastery of tessellation. At the same timwas dissatised with the way the patternarbitrarily interrupted at the edges.

would not permit what the real materials ofhis work shop required: a nite boundary. Hesought a new logic, explicitly visual, by whichhe could organize actually innite populationsof his corporeal motifs into a patch of nitearea. Within the framework of graphic art, hesought, he said, to capture innity.

SerendipityIn 1954, the organizing committee for theInternational Congress of Mathe maticianspromoted an unusual special event: an exhi-

bition of the work of Escher at the StedelijkMuseum in Amsterdam. In the companioncatalogue for the exhibition, the committeecalled attention not only to the mathemati-cal substance of Eschers tessellations butalso to their peculiar charm. Tree yearslater, while writing an article on symme-try to serve as the pres idential address tothe Royal Society of Canada, the eminentmathematician H.S.M. Coxeter recalled theexhibition. He wrote to Escher, requestingper mission to use two of his prints as illus-trations for the article. On June 21, 1957,Escher responded enthusiastically:

Not only am I willing to give you full permis-sion to pub lish reproductions of my regular-at-llings, but I am also proud of your inter-est in them!

In the spring of 1958, Coxeter sent to Eschera copy of the article he had written. In addi-tion to the prints of Eschers at-llings,the article contained the following gure,which we shall call Figure A:

Immediately, Escher saw in the gure arealistic method for achieving his goal: tocapture innity. For a suitable motif, such

as an angel or a devil, he might create, inmethod logically precise and in form visuallpleasing, innitely many modied copies othe motif, with the intended effect that themultitude would pack neatly into a disk.

With straightedge and compass, Escherset forth to analyze the gure. Te followingdiagram, based upon a workshop drawingsuggests his rst (no doubt empirical) effort

Workshop drawing

Escher recognized that the gure is denedby a network of innitely many circular arcs

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together with certain diameters, each of whichmeets the circular boundary of the ambientdisk at right angles. o reproduce the gure, heneeded to determine the centers and the radiiof the arcs. Of course, he recognized that thecenters lie exterior to the disk.

Failing to progress, Escher set the projectaside for several months. Ten, on Novem-ber 9, 1958, he wrote a hopeful letter to hisson George:

Im engrossed again in the study of an illus-tration which I came across in a publication ofthe Canadian professor H.S.M. Coxeter . . . Iam trying to glean from it a method for reduc-ing a plane-lling motif which goes from thecenter of a circle out to the edge, where themotifs will be innitely close together. Hishocus-pocus text is of no use to me at all,but the picture can probably help me to pro-duce a division of the plane which promisesto become an entirely new variation of myseries of divisions of the plane. A regular,circular division of the plane, logically bor-dered on all sides by the innitesimal, issomething truly beautiful.

Soon after, by a remarkable empirical effort,Escher succeeded in adapting Coxetersgure to serve as the underlying pattern forthe rst woodcut in his Circle Limit Series,CLI (November 1958).

One can detect the design for CLI in thefollowing Figure B, closely related to Figure A:

FrustrationHowever, Escher had not yet found the

principles of construction that un derlieFigures A and B. While he could reproducethe gures empirically, he could not yetconstruct them ab initio, nor could he con-struct variations of them. He sought Cox-eters help. What followed was a comedyof good inten tion and miscommunication.Te artist hoped for the particular, in prac-tical terms; the mathematician offered thegeneral, in esoteric terms. On December 5,1958, Escher wrote to Coxeter:

Though the text of your article on CrystalSymmetry and its Generalizations is muchtoo learned for a simple, self-made planepattern-man like me, some of the text illustra-tions and especially Figure 7, [that is, Figure A]gave me quite a shock.

Since a long time I am interested in pat-terns with motifs getting smaller and small-er till they reach the limit of innite smallness.The question is relatively simple if the limit isa point in the center of a pattern. Also, a line-limit is not new to me, but I was never able tomake a pattern in which each blot is gettingsmaller gradually from a center towards theoutside circle-limit, as shows your Figure 7.

I tried to nd out how this gure was geo-metrically con structed, but I succeeded only innding the centers and the radii of the largestinner circles (see enclosure). If you could give mea simple explanation how to construct the fol-lowing circles, whose centers approach graduallyfrom the outside till they reach the limit, I shouldbe immensely pleased and very thankful to you!Are there other systems besides this one to reacha circle-limit?

Nevertheless I used your model for a largewoodcut (CLI ), of which I executed only a sectorof 120 degrees in wood, which I printed three

Escher completed CLI, the rst in theCircle Limit Series, in 1958.

Regular Division VI (1957) illustrates Eschersability to execute a line limit.

Capturing Innity continued

Escher based the design ofCLIon Figure B,which he derived from Figure A.

The two are superimposed in Figure AB.

Figure B

Figure AB

times. I am sending you a copy of it, with another little one (Regular Division illustrating a line-limit case.

On December 29, 1958, Coxeter replied:

I am glad you like my Figure 7, anded that you succeeded in reconstrucmuch of the surrounding skeletonserves to locate the centers of the cThis can be continued in the same mFor instance, the point that I have mon your drawing (with a red on the bacthe page) lies on three of your circcenters 1, 4, 5. These centers therefore liestraight line (which I have drawn faintly and the fourth circle through the red poin

have its center on this same red line. In answer to your question Are thesystems be sides this one to reach a limit? I say yes, innitely many! Thular pattern [that is, Figure A] is den{4, 6} because there are 4 white and triangles coming together at some pand 6 at others. But such patterns {p,for all greater values of p and q and a= 3 and q = 7,8,9,... A different but retern, called is obtained by drawnew circles through the right anglewhere just 2 white and 2 shaded trcome together. I enclose a spare copy

7>> If you like this pattern with itstriangles and heptagons, you can easilfrom {4, 6} the analogue , wsists of squares and hexagons.

One may ask why Coxeter would senEscher a pattern featuring sevenfold symmetry, even if merely to serve as an analogySuch a pattern cannot be constructed withstraightedge and compass. It could onlycause confusion for Escher.

However, Coxeter did present, though



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very briey, the principle that Eschersought. I have displayed the essential sen-

tence in italics. In due course, I will showthat the sentence holds the key to decipher-ing Coxeters gure. Clearly, Escher did notunderstand its signicance at that time.

On February 15, 1959, Escher wroteagain, in frustration, to his son George:

Coxeters letter shows that an innite num-ber of other systems is possible and that,instead of the values 2 and 3, an innite num-ber of higher values can be used as a basis. Heencloses an example, using the values 3 and 7of all things! However, this odd 7 is no use tome at all; I long for 2 and 4 (or 4 and 8), because

I can use these to ll a plane in such a way thatall the animal gures whose body axes lie inthe same circle also have the same colour,whereas, in the other example (CLI), 2 whiteones and 2 black ones constantly alternate.My great enthusiasm for this sort of pictureand my tenacity in pursuing the study will per-haps lead to a satisfactory solution in the end.Although Coxeter could help me by saying justone word, I prefer to nd it myself for the timebeing, also because I am so often at cross pur-poses with those theoretical mathematicians,on a variety of points. In addition, it seems tobe very difcult for Coxeter to write intelligibly

for a layman. Finally, no matter how difcult itis, I feel all the more satisfaction from solving aproblem like this in my own bumbling fashion.But the sad and frustrating fact remains thatthese days Im starting to speak a languagewhich is understood by very few people. Itmakes me feel increasingly lonely. After all, Ino longer belong anywhere. The mathemati-cians may be friendly and interested and giveme a fatherly pat on the back, but in the endI am only a bungler to them. Artistic peoplemainly become irritated.

SuccessEschers enthusiasm and tenacity did indeed

prove sufficient. Somehow, dur ing the fol-lowing months, he taught himself, in termsof the straightedge and the compass, to con-struct not only Coxeters gure but at leastone variation of it as well. In March 1959,he completed the second of the woodcuts inhis Circle Limit Series.

he simplistic design of the work sug-gests that it may have served as a practicerun for its successors. In any case, Escherspoke of it in humorous terms:

Really, this version ought to be painted onthe inside sur face of a half-sphere. I offeredit to Pope Paul, so that he could decorate theinside of the cupola of St. Peters with it. Justimagine an innite number of crosses hang-ing over your head! But Paul didnt want it.

In December 1959, he completed thethird in the series, the intriguing CLIII, titledTe Miraculous Draught of Fishes.

He described the work eloquently, inwords that reveal the craftsmans pride ofachievement:

In the colored woodcut Circle Limit III theshortcomings of Circle Limit I are largely elimi-nated. We now have none but through trafcseries, and all the sh belonging to one serieshave the same color and swim after each otherhead to tail along a circular route from edgeto edge. The nearer they get to the center thelarger they become. Four colors are neededso that each row can be in complete contrastto its surroundings. As all these strings of shshoot up like rockets from the innite distanceat right angles from the boundary and fall backagain whence they came, not one single com-ponent reaches the edge. For beyond that there

Despite its simplistic motif,CLII (1959) represented anartistic breakthrough: Escher was now able to construct

variations of Coxeters gures.

Six months after his breakthrough withCLII,Escher produced the more sophisticatedCLIII,

The Miraculous Draught of Fishes.(1959).

Figure C

Figure D

Figure CD

is absolute nothingness. And yet thiworld cannot exist without the emparound it, not simply because withinposes without, but also because it is oin the nothingness that the center pothe arcs that go to build up the framewxed with such geometric exactitude. As I have noted, Escher completed the las

of the Circle Limit Series, CLIV, in July 1960Of this work, he wrote very little of substance

Here, too, we have the components dimin size as they move outwards. The six(three white angels and three black devarranged about the center and radiate fThe disc is divided into six sections iturn and turn about, the angels on abackground and then the devils on a whgain the upper hand. In this way, heavhell change place six times. In the interearthly stages, they are equivalent.

Perhaps Escher intended that this woodcut

should inspire not commentary but con-templation.Remarkably, while CLI and CLIV ar

based upon Figures A and B, CLII and CLIare based upon the following subtle variations of them:



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Straightedge and Compass

Let me describe how I myself wouldreconstruct the critical Figure A, withstraightedge and compass. Such anexercise might shed light on Eschersprocedures. To that end, I will suppressmy knowledge of mathematics beyondelementary geometry. However, at acertain point, I will allow myself to be,like Escher, preternaturally clever.

To begin, let me denote by H thedisk that serves as the foundation forthe gure. Moreover, let me declarethat the radius of H is simply one unit.I note that there are six diameters,separated in succession by angles of30 degrees, that emphasize the rota-tional symmetry of the gure. I alsonote that, among the circular arcsthat dene the gure, there are six forwhich the radii are largest. By roughmeasurement, I conjecture that theradii of these arcs equal the radius ofH and that the centers of the arcs lie 2units from the center ofH . I display myconjectures in the following diagram:

Step 1

The bold brown disk is H. Clearly,the six red circles meet the boundaryof H at right angles. By comparisonwith Figure A, I see that I am on theright track.

The diagram calls out for its ownelaboration. I note the points of inter-section of the six red circles. I drawthe line segments joining, in succes-sion, the centers of these circles and Imark the midpoints of the segments.Using these midpoints and the pointsof intersection just mentioned, I drawsix new circles. Then, from the newcircles, I do it all again. In the follow-ing gure, I display the results of mywork: the rst set of new circles ingreen; the second, in blue.

Step 2

Now the diagram falls mute. Isee that the blue circles offer no newpoints of intersection from whichto repeat my mechanical maneu-vers. Of course, the red circles andthe blue circles offer new points ofintersection, but it is not clear whatto do with them. Perhaps Escherencountered this obstacle, calledupon Coxeter for help, but thenretired to his workshop to confrontthe problem on his own. In any case,I must now nd the general princi-ples that underlie the construction,by straightedge and compass, of thecircles that meet the boundary of Hat right angles. I shall refer to thesecircles as hypercircles.

To that end, I propose the follow-ing diagram:

The Polar Construction

Again, the bold brown disk is H.The perpendicular white lines set theorientation for the construction. I con-tend that, from the red point or theblue point, I can proceed to constructthe entire diagram. In fact, from thered point, I can draw the white dogleg.From the blue point, I can draw the blue

circle. In either case, I can proceed byobvious steps to complete the diagram.Now, with the condence of experience,I declare that the red circle is a hyper-circle. Obviously, it meets the horizontalwhite diameter at right angles.

I shall refer to the foregoing con-struction as the Polar Construction. Inrelation to it, I shall require certain ter-minology. I shall refer to the red pointas the base point, to the blue point asthe polar point, and to the white pointas the point inverse to the base point.I shall refer to the red circle as thehypercircle, to the (vertical) red andblue lines as the base line and the polarline, respectively, and to the (horizon-tal) white line as the diameter.

By design, the polar constructionsand the hypercircles stand in perfectcorrespondence, each determiningthe other. However, to apply a polarconstruction to construct a particu-lar hypercircle passing through anarbitrary point, one must rst locatethe base point for the construction,that is, the point on the hypercirclethat lies closest to the center of H. Inpractice, that may be difcult to do. Irequire greater exibility.

By experimentation with the PolarConstruction, I discover the elegantPrinciple of Polar Lines:

If several hypercircles pass through acommon point then their centers mustlie on a common line, in fact, the polarline for the common point.

and a specialized but useful corollary,the Principle of Base Lines:

If two hypercircles meet at right anglesthen the center of the one must lie onthe base line of the other.

The following diagram illuboth principles:

Principles of Polar/Base Line

For the rst principle, themon point is the red base poipolar construction and the coline is the corresponding bluline. Moreover, the two greencircles pass through not onbase point but also the whiteinverse to it. Finally, in accothe facts of elementary geomthe angle of intersection bethe two hypercircles coincidthe angle between the two sponding green radii.

For the second principlorange base line for the lowercircle passes through the centeupper hypercircle.

At this point, I should notin his letter of December 29Coxeter offered the Principle Lines to Escher.

With the foregoing princimind, I return to the former pstagnation. I engage the diaas if in a game of chess. For apoint of intersection betweencircles offered by the diagramthe corresponding polar constion. I determine which amoother hypercircles passing ththe point are required for proApplying the Principle of PolI draw them. (Sometimes, thciple of Base Lines provides cut. Sometimes, good fortuna role. These elements lend a tapiquancy to the project.) ThaI look for new points of interoffered by the diagram: those by the new hypercircles thatdrawn. And so I continue, lessly, until I encounter a faimotor control, of visual acuitwillpower.

I present the following diwith a challenge: Justify the dof the orange and purple circ

Constructing the ScaffoldEscher wrote much about the designs for his regular divisions ofthe Euclidean plane, but nothing about the principles underlyingthe Circle Limit Series. He left only cryptic glimpses. From hisworkshop drawings, one can see that, in effect, he created ascaffold of lines in the nothingness exterior to the basic disk,

from which he could draw the circles that compose the desiredgure. However, one cannot determine with certainty how hefound his way. Did he reconsider Coxeters letter? Did he discover(by trial and error) and formulate (in precise terms) the principleswhich underlie the design of Figures A, B, C, and D? Lacking thecertain, I will offer the plausible.



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For instance, in Figure D and in CLIII, theeight vertices of the central octagon correspond,alternately, to threefold focal points of the nosesand the wings of the ying sh. Similarly, in Fig-ure B and in CLIV, the six vertices of the centralhexagon correspond to fourfold focal points ofthe wing tips of the angels and the devils.

Clearly, Escher had found and masteredhis new logic. Within the frame work ofgraphic art, by his own resources, he hadcaptured innity.

A Subjective ViewMathematicians cite CLIII as the most inter-esting of the woodcuts of the Circle LimitSeries. Tey enjoy especially the applicationof color, because it enriches the interpreta-tion of symmetry, and they are delighted bythe various implicit elements of surprise.Indeed, the redoubtable Coxeter calledattention to one such element, namely, thatthe white circular arcs in CLIII, which guidethe traffic ow of the ying sh, meet theboundary of the ambient disk not at rightangles but at angles of roughly 80 degrees,in contradiction with Eschers prior, rathermore poetic assertion. Coxeter wrote:

Eschers integrity is revealed in the fact that hedrew this angle correctly even though he appar-ently believed that it ought to be 90 degrees.

In my estimation, however, CLIV standsalone. It is the most mature of the woodcuts

of the Circle Limit Series. It inspires not activeanalysis but passive contemplation. It speaksnot in the brass tones of the cartoon but in thegold tones of the graceful and the grotesque.Like its relatives in the ornamental art ofthe Middle East, it prepares the mind of theobserver to see, in the local nite, hints of theglobal innite. It is, in fact, a beautiful visualsynthesis of Eschers meditation on innity:

We are incapable of imagining that time couldever stop. For us, even if the earth should ceaseturning on its axis and revolving around the sun,even if there were no longer days and nights,summers and winters, time would continue toow on eternally. We nd it impossible to imaginethat somewhere beyond the farthest stars of thenight sky there should come an end to space, afrontier beyond which there is nothing more . . .For this reason, as long as there have been men tolie and sit and stand upon this globe, or to crawland walk upon it, or to sail and ride and y acrossit, or to y away from it, we have held rmly to thenotion of a hereafter: a purgatory, heaven, hell,rebirth, and nirvana, all of which must continueto be everlasting in time and innite in space.


A challenge

The Circle Limit Series requiresrened ground plans, dened bylegions of hypercircles. In preparingthe plans, Escher gave new mean-ing to the words enthusiasmand tenacity.

To draw such a gure as Figure Aor Figure C, one must know where tobegin. In primitive terms, one must beable to construct the triangles at thecenters of the gures. For the case ofFigure A, the construction is simple.As described, one begins with a hyper-circle for which the radius is one unitand for which the center lies 2 unitsfrom the center of H. However, for thecase of Figure C, the construction ismore difcult. Of course, Escher musthave found a way to do it, since heused the gure as the ground plan for CLII andCLIII.

In any case, I have posted a suit-able construction on my website:people.reed.edu/~wieting/essays/HyperTriangles.pdf.

Perhaps it coincides with Eschersconstruction.

Detail of a challenge

Te Hyperbolic PlaneOn May 1, 1960, Escher sent a print of

CLIIIto Coxeter. Again, his words reveal hipride of achievement:

A minimum of four woodblocks, one color and a fth for the black lines, wasEvery block was roughly the form of aof 90 degrees. This im plicates that the compprint is composed of 4 x 5 = 20 printing

Responding on May 16, 1960, Coxeteexpressed thanks for the gift and ad mirationfor the print. Ten, in a virtuoso display ofinformed seeing , he described, mathematically, the mathematical elements implicit inCLIII, cit ing not only his own publicationsbut also W. Burnsides Teory of Groups fogood measure. For Coxeter, it was the ultimate act of respect. For Escher, however, itwas yet another encounter with the bafflingworld of mathematical abstraction. welvedays later, he wrote to George:

I had an enthusiastic letter from Cabout my colored sh, which I senThree pages of explanation of what I did. . . . Its a pity that I understand nabsolutely nothing of it.

One can only wonder at Coxeters insensitivity to the context of Eschers work: to thsteady applications of straightedge and compass; to the sound of the gouge on pearwoodand the smell of printers ink. Tat said, onecan only wonder at Eschers stubborn refusato explore what Coxeter offered: an invitation to the hyperbolic plane.

Let me elaborate. For more than two mil-lennia, the ve postulates of Euclid had governed the study of plane geometry. Te rstthree postulates were homespun rules thatactivated the straightedge and the compass.he fourth and fth postulates were moresophisticated rules that entailed the funda-mental Principle of Parallels, characteristiof Euclidean geometry:

For any point P and for any straight P does not lie on L then there isprecisely ostraight line M such that P lies on M athat L and M are parallel.

Specically, the fourth postulate entailedthe existence of the parallel M and the fthpostulate entailed the uniqueness.

he following diagram illustrates thePrinciple of Parallels in Euclidean geometry.Te rectangle E represents the conventionalmodel of the Euclidean plane: a perfectly atdrawing board that extends, in our imagina-tion, indenitely in all directions. Te pointP and the straight line L appear in red. Testraight line M appears in blue.



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Euclidean parallels

In the beginning, all mathematiciansregarded the postulates of Euclid as incontro-vertibly true. However, they observed that thefth postulate offered nothing constructiveand they believed that it was redundant. Teysought to prove the fth postulate from therst four. In effect, they sought to prove thatthe existence of the parallel M entailed its ownuniqueness. o that end, they applied the mostexible of the logicians methods: reductio adabsurdum. Tey supposed that the fth postu-late was false and they sought to derive fromthat supposition (together, of course, with therst four postulates) a contradiction. Succeed-ing, they would conclude that the fth postu-late followed from the rst four. For more thantwo millennia, many sought and all failed.

At the turn of the 18th century, the gripof belief in the incontro vertible truth ofthe fth postulate began to weaken. Manymathematicians came to believe that thesought contradiction did not exist. heycame to regard the propositions that theyhad proved from the negation of the fth

postulate not as absurdities leading ulti-mately to a presumed contradiction but asprovocative elements of a new geometry.

Swiftly, the new geometry acquired dis-ciples, notably, the young Russian mathemati-cian N. Lobachevsky and the young Hungar-ian mathematician J. Bolyai. Tey and manyothers proved startling propositions at vari-ance with the familiar propositions of Euclid-ean geometry. Te German savant K. Gausshad pondered these matters for 30 years. In1824, he wrote to his friend F. aurinus:

The theorems of this geometry appear to beparadoxical and, to the uninitiated, absurd;but calm, steady reection reveals that theycontain nothing at all impossible. For exam-ple, the three angles of a triangle becomeas small as one wishes, if only the sides aretaken large enough; and the area of a trianglecan never exceed a denite limit.

However, the specter of contradiction, oncesought by all but now by many feared, contin-ued to cast its shadow over the planes. Fifty years would pass before mathematicians found

a method by which they could, decisively, ban-ish the specter: the method of models.

Let me explain the method in terms of acase study. At the turn of the 19th century,the French savant H. Poincar suggested anovel interpretation of the points and thestraight lines of the new geometry, using theelements of the Euclidean plane E itself. Hedeclared that the points of the new geom-etry shall be interpreted as the points ofthe unit disk H, the same disk that would,in due course, serve Escher in his plans forthe Circle Limit Series. He declared that thestraight lines of the new geometry shall beinterpreted as the arcs of circles that meetthe boundary of H at right angles.

When I use a word, Humpty Dumpty said inrather a scornful tone, it means just what Ichoose it to meanneither more nor less.

Tese interpretations can be justied, in asense, by introducing an un usual methodfor measuring distance between points inH, with respect to which the shortest pathsbetween points prove to be, in fact, subarcsof arcs of the sort just described. Moreover,the lengths of the various straight linesprove to be innite. Te same is true of thearea of H.

Poincar then proved that H servedas a model for the new geometry. Tat is,he proved that the rst four postulates ofEuclid are true in H and the fth postulate isfalse. He concluded that if, by a certain argu-ment, one should nd a contradiction in thenew geometry, then, by the same argument,one would nd a contradiction in Euclideangeometry as well. In turn, he concluded thatif Euclidean geometry is free of contradic-tion, then the new geometry is also free ofcontradiction.

By similar (though somewhat more sub-tle) maneuvers, one can show the converse:if the new geometry is free of contradiction,then Euclidean geometry is also free of con-tradiction.

he following diagram illustrates the

Principle of Parallels in the new geometry:For any point P and for any straight line L, if Pdoes not lie on L then there are many straightlines M such that P lies on M and such that Land M are parallel.

Te disk H represents the model of thehyperbolic plane designed by Poincar. Tepoint P and the straight line L appear in red. Various parallels M appear in blue while thetwo parallels that meet L at innity appearin green.

Hyperbolic parallels

After more than two millennia of contentions to the contrary, we now know that theEuclidean plane is not the only rationallycompelling context for the study of plangeometry. From a logical point of view, thEuclidean geometry and the new geometrycalled hyperbolic, are equally tenable.

In light of the foregoing elaboration, can set Eschers Circle Limit Series perspective by describing the strik incontrast between regular tessellations

of the Euclidean plane and regular tessellations of the hyperbolic plane. Of the formerthere are just three instances: the tessella-tion , dened by the regular 3-gon (thatis, the equi lateral triangle); the tessellatioH, dened by the regular 6-gon (that is, theregular hexagon); and the tessellation Sdened by the regular 4-gon (better knownas the square). Tese are the ground forms forall tessellations of the Euclidean plane. Tetessellations and H are mutually dual, inthe sense that each determines the other by

drawing line segments between midpoints ofcells. In that same sense, the tessellation S iself-dual. In the following gures, I displthe tessellations and H superimposed, andthe tessellation S in calm isolation:


Figure TH

Figure S

E

H



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Of the hyperbolic plane, however, thereare innitely many tessellations, with prop-erties that defy visualization. Indeed, forany positive integers p and q for which(p - 2)(q - 2) exceeds 4, there is a regular tes-sellation, called (p,q), by regular p-gons, q ofwhich turn about each vertex. Te followingtwo illustrations suggest the superposition,Figure B, of the mutually dual tessellations

(4,6) and (6,4) and the superposition, FigureD, of the mutually dual tessellations (3,8)and (8,3). One can see that these are thetessellations that served as Eschers groundplans for the Circle Limit Series:

In the rst gure, one nds regular 4-gons(in red), six of which turn about each vertex;and regular 6-gons (in blue), four of which turnabout each vertex. In the second gure, onends regular 3-gons (in red), eight of whichturn about each vertex; and regular 8-gons (inblue), three of which turn about each vertex.

For the regular tessellations of the Euclid-ean plane, the various cells of a given color are,plain to see, mutually congruent. Remarkably,for the regular tessellations of the hyperbol-ic plane, the same is true. Of course, to the

Euclidean eye, the latter assertion would seemto be wildly false. However, to the hyperboliceye, conditioned to the unusual method ofmeasuring distance, the assertion is true.

Of course, the assertion of congruenceapplies just as well to the various motifs thatcompose the patterns of the Circle LimitSeries. Although there is no evidence thatEscher understood this assertion, I am surethat he would have been delighted by theidea of a hyperbolic eye that would conrm his

procedure for capturing innity and wouldrene its meaning.

ConclusionSeeking a new visual logic by which tocapture innity, Escher stepped, withoutforeknowledge, from the Euclidean planeto the hyperbolic plane. Of the former, hewas the master; in the latter, a novice. Nev-ertheless, his acquired insights yielded twoamong his most interesting works: CLIII,he Miraculous Draught of Fishes, and CLIV,

Angels and Devils. Escher devoted 25 years of his life to the

development of strik ing, perplexing imagesand patterns: those that so fascinated himthat he felt driven to communicate them toothers. In retrospect, it seems to me alto-gether tting and proper that non-Euclideangeometry should have served, at least implic-itly, as the inspiration for his later works.

CodaIn my imagination, I see the crystal spheresof Art and Mathematics rotating rapidlyabout their axes and revolving slowly abouttheir center of mass, in the pure aethersurrounding them. I see ribbons of lightash between them and within these thereections, the cryptic images of diamantineforms sparkle and shimmer. As if in a dream,I try to decipher the images: simply, deeplyto understand.

Acknowledgements: The several graphics worksCLICLII, CLIII, CLIV, Day and Night, Regular Division IIIDivision VI,and Hand with Reflectings Sphere) and thephotographs of Escher are printed here by permof the M.C. Escher Company-Holland 2010. Areserved. www.mcescher.com. The line illustratmy own, though I must confess that I made themwith Eschers tools, the straightedge and the com

but with the graphics subroutines which figure iomnipresent computer program Mathematica, inby the symmetries of the hyperbolic plane. For source of the Workshop Drawing, I am indebtedD. Schattschneider. The excerpts of correspondebetween Escher and Coxeter are drawn from theof the National Gallery of Art. The excerpts of lfrom Escher to his son George are drawn from tby H. Bool,M.C. Escher: His Life and Complete Graphic 1981. The several excerpts of Eschers essays afrom two books by Escher, M.C. Escher: the Graphic Wo 1959, andEscher on Escher: Exploring the Infinite, 1989; afrom the book by B. Ernst,The Magic Mirror of M.C. Es1984. The papers by Coxeter appear in the Tranof the Royal Society of Canada, 1957, and in LeVolume 12, 1979. The excerpt of the letter fromto F. Taurinus appears in the book by M. GreenbEuclidean and NonEuclidean Geometries,1980.course, the pronouncement by Humpty-Dumptyin the book by L. Carroll,Through the Looking-Glass, 1904The language of the Coda carries, intentionally, echo of the beautiful reverie with which Escherto a close his essay, Voyage to Canada. The Codexpresses, from the heart, my metaphor for the between the magisteria of Art and Mathematics.I am indebted to C. Lydgate, the editor ofReed, for hisencouragement during the preparation of this esfor his many useful suggestions for improvemen

About the Author: Professor Wieting receivedthe degree of Bachelor of Science in Mathematicfrom Washington and Lee University in 1960 anddegree of Doctor of Philosophy in Mathematics fHarvard University in 1973. He joined the mathefaculty at Reed in 1965. His research interests incrystallography, cosmology, and ornamental art. Wieting draws inspiration from Chaucers descrithe Clerke:Gladley wolde he lerne and gladley teche.

Figure B

Figure D

Hand with Reecting Sphere(1935)

Escher contemplatesAngels and Devils in his study.


capturing infinity

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