Escher's Metaphors

download Escher's Metaphors

of 6

Transcript of Escher's Metaphors

  • 8/12/2019 Escher's Metaphors

    1/6

    SCIENCE IN PICTURES

    Eschers MetaphorsThe prints and drawings of M.C. Escher

    give expression to abstract conceptsof mathematics and science

    by Doris Schattschneider

    Mbius Strip II, 1963

    66 SCIENTIFIC AMERICAN November 1994 Copyright 1994 Scientific American, Inc.

  • 8/12/2019 Escher's Metaphors

    2/6

    Throughout his life Mauritsnelis Escher (he used only M.C.)remarked on his inability to

    understand mathematics, declaringhimself absolutely innocent oftraining or knowledge in the ex-act sciences. Yet even as a child,Escher was intrigued by orderand symmetry. The fascinationlater led him to study patterns

    of tiles at the Alhambra in Gra-nada, to look at geometric draw-ings in mathematical papers (withthe advice of his geologist broth-er) and ultimately to pursue hisown unique ideas for tiling a plane.

    Eschers attention to the coloring ofhis drawings of interlocked tiles anticipat-ed the later work of mathematicians and crys-tallographers in the eld of color symmetry. Hisworks are now commonly used to illustrate these concepts.His exhibit in conjunction with the 1954 International Con-gress of Mathematicians in Amsterdam and the publication

    of his rst book (The GraphicWork of M. C. Escher) in 1959struck a chord with mathe-maticians and scientists thatstill resonates strongly. He

    wrote that a main impetusfor his work was a keen in-

    terest in the geometric lawscontained by nature around us.

    In expressing his ideas in graphicworks, he provided arresting visual

    metaphors for fundamental ideas inscience.

    Escher was born in 1898 in the town ofLeeuwarden, Holland. The youngest son of a civ-

    il engineer, he grew up with four brothers in Arnhem. Al-though three of his brothers pursued science or engineering,Escher was a poor mathematics student. With the encour-

    M. C. ESCHER views himself in a mir-ror in this scratch drawing with

    lithographic ink.

    Cor-

    Self-Portrait, 1943

    MOBIUS STRIP II harbors a procession of ants crawling in anendless cycle. With a nite number of gures, Escher de-

    picts innity through the continuous traversal of anendless loop. The ants demonstrate as well that

    this unusual loop (originally printed vertically)has only one side.

    SCIENTIFIC AMERICAN November 1994 67Copyright 1994 Scientific American, Inc.

  • 8/12/2019 Escher's Metaphors

    3/6

    agement of his high school art teacher,he became interested in graphic arts,rst making linoleum cuts.

    In 1919 he entered the School for Ar-chitecture and Decorative Arts in Haar-lem, intending to study architecture. Butwhen he showed his work to Samuel Jes-surun de Mesquita, who taught graphicarts there, he was invited to concen-trate in that eld. De Mesquita had a

    profound inuence on Escher, both asa teacher (particularly of woodcut tech-niques) and later as a friend and fellowartist.

    After nishing his studies in Haarlem,Escher settled in Rome and made manyextensive sketching tours, mostly insouthern Italy. His eyes discerned strik-ing visual eects in the ordinaryar-chitectural details of monumental build-ings from unusual vantage points, lightand shadow cast by warrens of stair-cases in tiny villages, clusters of hous-es clinging to mountain slopes that

    plunged to distant valleys and, at theopposite scale, tiny details of nature asif viewed through a magnifying glass.In his studio, he would transform thesketches into woodcuts and lithographs.

    In 1935 the political situation becameunendurable, and with his wife andyoung sons, Escher left Italy forever. Af-ter two years in Switzerland and thenthree years in Uccle, near Brussels, they

    settled permanently in Baarn, Holland.These years also brought an abrupt turnin Eschers work. Almost all of it fromthis time on would draw its inspirationnot from what his eyes observed butrather from his minds eye. He sought togive visual expression to concepts andto portray the ambiguities of human ob-servation and understanding. In do-ing so, he often found himself in a

    world governed by mathematics.Escher was fascinated, almost

    obsessed, with the concept of theregular division of the plane.In his lifetime, he producedmore than 150 color drawingsthat testied to his ingenu-ity in creating gures thatcrawled, swam and soared,yet lled the plane with theirclones. These drawings illus-trate symmetries of manydierent kinds. But for Esch-er, division of the plane was

    also a means of capturing in-nity. Although a tiling suchas the one using butteries[see illustration below] can inprinciple be continued indef-initely, thus giving a sugges-tion of innity, Escher was chal-lenged to contain innity with-in the connes of a single page.

    Anyone who plunges into inn-

    Triangle System I B3Type 2, 1948

    SYMMETRY is a structural concept thatshapes many mathematical and physi-cal models. In Eschers drawing the

    butteries seem to ll the page ran-domly, yet each one is precisely placed

    and surrounded in exactly the sameway. Always six (in alternating colors)swirl about a point where left frontwingtips meet; always three (in differ-ent colors) spin about a point whereright back wings touch; and alwayspairs (of different colors) line up theedges of their right front wings. Alongwith rotational symmetry, the draw-ing has translational symmetry basedon a triangular grid. The pattern cancontinue forever in all directions andso provides an implicit metaphor ofinnity. Eschers attention to coloringanticipated discoveries by mathemati-

    cians in the eld of color symmetry.

    Circle Limit IV,1960

    Copyright 1994 Scientific American, Inc.

  • 8/12/2019 Escher's Metaphors

    4/6

    ity, in both time and space, far-ther and farther without stop-ping, needs xed points, mile-posts as he ashes by, for oth-erwise his movement is indis-tinguishable from standingstill, Escher wrote. He mustmark o his universe intounits of a certain length,into compartments whichrepeat one another in end-less succession.

    After completing several

    prints in which gures end-lessly diminish in size asthey approach a central van-ishing point [see Whirlpoolson page 71], Escher sought a

    device to portray progressivereduction in the opposite direc-

    tion. He wanted gures that re-peated forever, always approach-

    ingyet never reachingan encir-

    cling boundary. In 1957 the mathema-tician H.S.M. Coxeter sent Escher a re-print of a journal article in which he il-lustrated planar symmetry with someof Eschers drawings. There Escherfound a gure that gave him quite ashocka hyperbolic tessellation of tri-

    DUALITY is perhaps the most prevalent theme in Escher's later prints. In mathematics, astatement has a negation, and a set has a complement; in every case, the object and itsdual completely define each other. In Circle Limit IV, there are no outlines. The contoursof the angels and devils dene one another. Either is gure or ground (Escher reminds us

    by omitting detail in half the gures). In this hyperbolic tiling the gures appear to our Eu-clidean eyes to become more distorted as they diminish in size. Yet measured by the ge-

    ometry intrinsic to the world of the print, every angel is exactly the same size andshape, and so is every devil. An innite number of copies repeat forever, never

    leaving the connes of the circle.

    SELF-SIMILARITY is illustrated in theprint Square Limit, constructed usinga recursive scheme of Eschers own in-vention. A set of directions that is ap-plied to an object to produce new ob-

    jects, then applied to the new objectsand so on, ad innitum, is called a re-cursive algorithm. The end product isself-similar when all the nal objectsare the same as the original, except forchanges of scale, orientation or posi-tion. A sketch (top right) sent by Esch-er to the mathematician H.S.M. Cox-eter to explain the print shows that theunderlying grid involves a recursivesplitting of isosceles triangles. In exe-cuting the print, Escher carved thewoodblock only for a triangle havingits apex at the center of the square andits base as one side of the squareand

    printed the block four times.

    Square Limit, 1964

    Copyright 1994 Scientific American, Inc.

  • 8/12/2019 Escher's Metaphors

    5/6

    angles that showed exactly the eect he sought. From a careful study

    of the diagram, Escher discerned the rules of tiling in which circulararcs meet the edge of an encompassing circle at right angles. Duringthe next three years, he produced four dierent prints based on thistype of grid, of which Circle Limit IV[see top illustration on precedingtwo pages] was the last.

    Four years later Escher devised his own solution to the problem ofinnity within a rectangle [see bottom illustration on preceding page].His recursive algorithma set of directions repeatedly applied to anobjectresults in a self-similar pattern in which each element is relat-ed to another by a change of scale. Escher sent Coxeter a sketch of the

    underlying grid, apologizing: I fear that the subject wont be very in-teresting, seen from your mathematical point of view, because its re-ally simple as a at lling. None the less it was a headaching job to ndan adequate method to realise the subject in the simplest possible way.

    In a lecture a few summers ago mathematician William P. Thurston,director of the Mathematical Sciences Research Institute at the Uni-versity of California at Berkeley, illustrated the concept of self-similartiling with just such a grid, unaware of Eschers earlier discovery.

    Curiously, self-similar patterns provide examples of gures that

    70 SCIENTIFIC AMERICAN November 1994

    RELATIVITY states that what an observer sees is inuenced by con-text and vantage point. In the lithograph High and Low, Escher pre-sents two dierent views of the same scene. In the lower half theviewer is on the patio; in the upper half the viewer is looking down.Now draw back from the print : Is that tiled diamond at the center ofthe print a oor or a ceiling? Escher uses it for both in order to marrythe two views. It is impossible to see the entire print in a logical way.The scene also illustrates how pasting local views together to form aglobal whole can lead to contradictions.

    DIMENSION is that concept whichclearly separates point, line, planeand space. To illustrate the ambi-guities in the perception of dimen-sion, Escher exploited the printedpagewhich always must fool theviewer when it depicts a three-dimensional scene. In Day andNight, the at checkerboard of

    farmland at the bottom of the printmetamorphoses into two ocks ofgeese. The print also illustratesthe concept of topological change,in which a gure is deformedwithout being cut or pierced. Re-ection and duality are present aswell: black geese y over a sunlitvillage, whereas white ones wingover a night view of a mirror im-age of the same scene.

    DORIS SCHATTSCHNEIDER, professor of mathematics at Moravian Col-lege in Bethlehem, Penn., received her Ph.D. from Yale University in 1966.Active as a teacher, writer and lecturer, the former editor of MathematicsMagazine has strong interests in both geometry and art. She is co-author(with Wallace Walker) of a popular work with geometric models, M.C. Esch-er Kaleidocycles. Her 1990 book, Visions of Symmetry, culminated a re-search project on the symmetry studies of M.C. Escher.

    Day and Night, 1938

    High and Low, 1947

    Copyright 1994 Scientific American, Inc.

  • 8/12/2019 Escher's Metaphors

    6/6

    SCIENTIFIC AMERICAN November 1994 71

    have fractional, or fractal, dimension, an ambiguity that Escherwould doubtless have enjoyed. In 1965 he confessed: I cannothelp mocking all our unwavering certainties. It is, for example,great fun deliberately to confuse two and three dimensions, theplane and space, and to poke fun at gravity. Escher was masterful

    at confusing dimensions, as in Day and Night [see upper illustra-tion on opposite page], in which two-dimensional farm elds mys-teriously metamorphose into three-dimensional geese. He also de-lighted in pointing out the ambiguities and contradictions inherentin a common practice of science: pasting together several localviews of an object to form a global whole [see lower illustration onopposite page].

    Near the end of his life (he died in 1972), Escher wrote, Aboveall, I am happy about the contact and friendship of mathemati-cians that resulted from it all. They have often given me new ideas,and sometimes there is even an interaction between us. How play-ful they can be, these learned ladies and gentlemen!

    INFINITY is conned withinthe nite space of a printin Whirlpools. The artistdraws a at projection of thecurve (a loxodrome) that istraced out on the globe by apath that cuts across all me-ridians at a constant angle.As any navigator knows, sail-ing such a rhumb line re-sults in a never-ending, ever

    tightening spiral about theearths pole. Escher usedone woodblock for both col-ors. Printing once for the red,he turned it halfway aroundand printed for the gray.

    REFLECTION allows phenom-ena to be observed that aretoo small, too far away or tooobscure to be seen directly.Puddle directs our eyes to awoodland trail imprinted by

    boots and tiresyet in thepuddle are also revealed sil-houetted trees arching over-head against a moonlit sky.Escher reminds us of the un-

    seen worlds below, behindand above our limited gaze.

    FURTHER READING

    THE GRAPHIC WORK OF M. C. ESCHER. M. C. Escher. Ballantine Books,1971.

    THE MAGIC MIRROR OF M. C. ESCHER. B. Ernst. Random House, 1976.ANGELS AND DEVILS. H.S.M. Coxeter in The Mathematical Gardner.Edited by David A. Klarner. Prindle, Weber and Schmidt, 1981.

    M. C. ESCHER: HIS LIFE AND COMPLETE GRAPHIC WORK. Edited by J. L.Locher. Harry N. Abrams, 1982.

    M. C. ESCHER: ART AND SCIENCE. Edited by H.S.M. Coxeter, M. Emmer, R.Penrose and M. L. Teuber. North-Holland, 1986.

    ESCHER ON ESCHER: EXPLORING THE INFINITE. M. C. Escher. Translatedby Karin Ford. Harry N. Abrams, 1989.VISIONS OF SYMMETRY: NOTEBOOKS, PERIODIC DRAWINGS, AND RELATEDWORK OF M. C. ESCHER. D. Schattschneider. W. H. Freeman and Com-pany, 1990.

    Whirlpools, 1957

    Puddle, 1952

    Copyright 1994 Scientific American, Inc.