# Capturing Infinity continued

In light of the foregoing elaboration, I can set Escher’s Circle Limit Series in perspective by describing the strik­ing contrast between regular tessellations of the Euclidean plane and regular tessellations of the hyperbolic plane. Of the former, there are just three instances: the tessellation T, defined by the regular 3-gon (that is, the equi­lateral triangle); the tessellation H, defined by the regular 6-gon (that is, the regular hexagon); and the tessellation S, defined by the regular 4-gon (better known as the square). These are the ground forms for all tessellations of the Euclidean plane. The tessellations T and H are mutually “dual,” in the sense that each determines the other by drawing line segments between midpoints of cells. In that same sense, the tessellation S is “self-dual.” In the following figures, I display the tessellations T and H superimposed, and the tessellation S in calm isolation:

Figure TH

Figure S

Of the hyperbolic plane, however, there are infinitely many tessellations, with properties that defy visualization. Indeed, for any positive integers p and q for which (p - 2)(q - 2) exceeds 4, there is a regular tessellation, called (p,q), by regular p-gons, q of which turn about each vertex. The following two illustrations suggest the superposition, Figure B, of the mutually dual tessellations (4,6) and (6,4) and the superposition, Figure D, of the mutually dual tessellations (3,8) and (8,3). One can see that these are the tessellations that served as Escher’s ground plans for the Circle Limit Series:

In the first figure, one finds regular 4-gons (in red), six of which turn about each vertex; and regular 6-gons (in blue), four of which turn about each vertex. In the second figure, one finds regular 3-gons (in red), eight of which turn about each vertex; and regular 8-gons (in blue), three of which turn about each vertex.

For the regular tessellations of the Euclidean plane, the various cells of a given color are, plain to see, mutually congruent. Remarkably, for the regular tessellations of the hyperbolic plane, the same is true. Of course, to the Euclidean eye, the latter assertion would seem to be wildly false. However, to the hyperbolic eye, conditioned to the “unusual method” of measuring distance, the assertion is true.

Of course, the assertion of congruence applies just as well to the various motifs that compose the patterns of the Circle Limit Series. Although there is no evidence that Escher understood this assertion, I am sure that he would have been delighted by the idea of a hyperbolic eye that would confirm his procedure for capturing infinity and would refine its meaning.

###### Conclusion

Seeking a new visual logic by which to “capture infinity,” Escher stepped, without foreknowledge, from the Euclidean plane to the hyperbolic plane. Of the former, he was the master; in the latter, a novice. Nevertheless, his acquired insights yielded two among his most interesting works: CLIII, The Miraculous Draught of Fishes, and CLIV, Angels and Devils.

Escher devoted 25 years of his life to the development of strik­ing, perplexing images and patterns: those that so fascinated him that he “felt driven to communicate them, to others.” In retrospect, it seems to me altogether fitting and proper that non-Euclidean geometry should have served, at least implicitly, as the inspiration for his later works.

###### Coda

In my imagination, I see the crystal spheres of Art and Mathematics rotating rapidly about their axes and revolving slowly about their center of mass, in the pure aether surrounding them. I see ribbons of light flash between them and within these the reflections, the cryptic images of diamantine forms sparkle and shimmer. As if in a dream, I try to decipher the images: simply, deeply to understand.

# Acknowledgements:

Escher contemplates Angels and Devils in his study.

The several graphics works (CLI, CLII, CLIII, CLIV, Day and Night, Regular Division III, Regular Division VI, and Hand with Reflectings Sphere) and the photographs of Escher are printed here by permission of the M.C. Escher Company-Holland © 2010. All rights reserved. www.mcescher.com. The line illustrations are my own, though I must confess that I made them not with Escher’s tools, the straightedge and the compass, but with the graphics subroutines which figure in the omnipresent computer program Mathematica, informed by the symmetries of the hyperbolic plane.  For the source of the Workshop Drawing, I am indebted to D. Schattschneider. The excerpts of correspondence between Escher and Coxeter are drawn from the Archives of the National Gallery of Art. The excerpts of letters from Escher to his son George are drawn from the book by H. Bool, M.C. Escher: His Life and Complete Graphic Work, 1981. The several excerpts of Escher’s essays are drawn from two books by Escher, M.C. Escher: the Graphic Work, 1959, and Escher on Escher: Exploring the Infinite, 1989; and from the book by B. Ernst, The Magic Mirror of M.C. Escher, 1984. The papers by Coxeter appear in the Transactions of the Royal Society of Canada, 1957, and in Leonardo, Volume 12, 1979. The excerpt of the letter from K. Gauss to F. Taurinus appears in the book by M. Greenberg, Euclidean and NonEuclidean Geometries, 1980.  Of course, the pronouncement by Humpty-Dumpty appears in the book by L. Carroll, Through the Looking-Glass, 1904. The language of the Coda carries, intentionally, a faint echo of the beautiful reverie with which Escher brings to a close his essay, Voyage to Canada. The Coda itself expresses, from the heart, my metaphor for the relation between the magisteria of Art and Mathematics. Finally, I am indebted to C. Lydgate, the editor of Reed, for his encouragement during the preparation of this essay and for his many useful suggestions for improvement.