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Feature Story March 2010 

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Capturing Infinity continuedPerhaps Escher intended that this woodcut should inspire not commentary but contemplation. Remarkably, while CLI and CLIV are based upon Figures A and B, CLII and CLIII are based upon the following subtle variations of them: Figure C Figure D Figure CD For instance, in Figure D and in CLIII, the eight vertices of the central octagon correspond, alternately, to threefold focal points of the noses and the wings of the flying fish. Similarly, in Figure B and in CLIV, the six vertices of the central hexagon correspond to fourfold focal points of the wing tips of the angels and the devils. Clearly, Escher had found and mastered his new logic. Within the framework of graphic art, by his own resources, he had captured infinity. A Subjective ViewMathematicians cite CLIII as the most interesting of the woodcuts of the Circle Limit Series. They enjoy especially the application of color, because it enriches the interpretation of symmetry, and they are delighted by the various implicit elements of surprise. Indeed, the redoubtable Coxeter called attention to one such element, namely, that the white circular arcs in CLIII, which guide the “traffic flow” of the flying fish, meet the boundary of the ambient disk not at right angles but at angles of roughly 80 degrees, in contradiction with Escher’s prior, rather more poetic assertion. Coxeter wrote: Escher’s integrity is revealed in the fact that he drew this angle correctly even though he apparently believed that it ought to be 90 degrees. In my estimation, however, CLIV stands alone. It is the most mature of the woodcuts of the Circle Limit Series. It inspires not active analysis but passive contemplation. It speaks not in the brass tones of the cartoon but in the gold tones of the graceful and the grotesque. Like its relatives in the ornamental art of the Middle East, it prepares the mind of the observer to see, in the local finite, hints of the global infinite. It is, in fact, a beautiful visual synthesis of Escher’s meditation on infinity: We are incapable of imagining that time could ever stop. For us, even if the earth should cease turning on its axis and revolving around the sun, even if there were no longer days and nights, summers and winters, time would continue to flow on eternally. We find it impossible to imagine that somewhere beyond the farthest stars of the night sky there should come an end to space, a frontier beyond which there is nothing more . . . For this reason, as long as there have been men to lie and sit and stand upon this globe, or to crawl and walk upon it, or to sail and ride and fly across it, or to fly away from it, we have held firmly to the notion of a hereafter: a purgatory, heaven, hell, rebirth, and nirvana, all of which must continue to be everlasting in time and infinite in space. The Hyperbolic PlaneOn May 1, 1960, Escher sent a print of CLIII to Coxeter. Again, his words reveal his pride of achievement: A minimum of four woodblocks, one for every color and a fifth for the black lines, was needed. Every block was roughly the form of a segment of 90 degrees. This implicates that the complete print is composed of 4 x 5 = 20 printings. Responding on May 16, 1960, Coxeter expressed thanks for the gift and admiration for the print. Then, in a virtuoso display of informed seeing, he described, mathematically, the mathematical elements implicit in CLIII, citing not only his own publications but also W. Burnside’s Theory of Groups for good measure. For Coxeter, it was the ultimate act of respect. For Escher, however, it was yet another encounter with the baffling world of mathematical abstraction. Twelve days later, he wrote to George: I had an enthusiastic letter from Coxeter about my colored fish, which I sent him. Three pages of explanation of what I actually did. . . . It’s a pity that I understand nothing, absolutely nothing of it. One can only wonder at Coxeter’s insensitivity to the context of Escher’s work: to the steady applications of straightedge and compass; to the sound of the gouge on pearwood and the smell of printer’s ink. That said, one can only wonder at Escher’s stubborn refusal to explore what Coxeter offered: an invitation to the hyperbolic plane. Let me elaborate. For more than two millennia, the five postulates of Euclid had governed the study of plane geometry. The first three postulates were homespun rules that activated the straightedge and the compass. The fourth and fifth postulates were more sophisticated rules that entailed the fundamental Principle of Parallels, characteristic of Euclidean geometry: For any point P and for any straight line L, if P does not lie on L then there is precisely one straight line M such that P lies on M and such that L and M are parallel. Specifically, the fourth postulate entailed the existence of the parallel M and the fifth postulate entailed the uniqueness. 
table of contents>Capturing Infinity  < Back12345678Next > 
March 2010 