@MISC{Goodman-Strauss01compassand, author = {Chaim Goodman-Strauss}, title = {Compass and Straightedge in the Poincaré Disk}, year = {2001} }

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Abstract

The spirit of this article belongs to another age. Today, “geometry” is most often analytic; this is especially suited to the making of nice pictures by computer. But here we give a synthetic approach to the development of hyperbolic geometry; our constructions use only a Euclidean compass and Euclidean straightedge, and can be carried out by hand. Indeed, M.C. Escher used something like the methods we give here to produce his well known Circle Limit I, II, III, and IV prints [9]. In [6], H.S.M. Coxeter describes a remarkable correspondence with Escher. Having met at the 1954 International Congress of Mathematics in Amsterdam, Coxeter apparently sent Escher a paper in which a drawing of part of a tiling of the Poincaré disk appeared. Coxeter must have been quite pleased and surprised to find a print of Circle Limit I in his mail in December 1958. It is quite remarkable that the drawing in the paper Coxeter had sent Escher was not even as detailed as our Figure 1, and did not show the “scaffolding ” Coxeter had used in its construction. Nonetheless Escher deduced and generalized the technique of its construction, producing incredibly fine tesselations of the Poincarédisk.