Compass and Straightedge in the Poincaré Disk
Compass and Straightedge in the Poincaré Disk
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DOI：
10.1080/00029890.2001.11919719
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发表时间：
2001-01
期刊：
影响因子：
--
通讯作者：
C. Goodman-Strauss
中科院分区：
文献类型：
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作者：
C. Goodman-Strauss
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 Poincare 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 Poincare disk. The technique for these constructions is quite ingenious and makes use of a certain duality between circular arcs and points inside the disk and points and Euclidean lines outside the disk, described at the end of Section 1. In [6], Coxeter only incidentally describes this duality; this is not quite enough to complete the construction and here we describe a full suite of available techniques. I believe nothing in this article can possibly be original: surely this was all wellknown at the end of the nineteenth century just as it has long been forgotten at the dawn of the twenty-first. There are closely related figures in the works of Gauss, Beltrami, Poincare, Schwarz, Klein, Fricke, Burnside, and others [16]. These sources are analytic or simply have other concerns; none, apparently, describe how these figures were produced. "Scaffolding" such as that of Figure 1 was shown only rarely, but it did appear and indeed, it seems to have been rediscovered several times. It seems reasonable to assume that many, perhaps nearly all, of the illustrated tilings of the Poincare disk were drafted, synthetically, by techniques very similar to those presented here. Remarkably, this may be the first detailed, explicit synthetic construction of triangle tilings of the Poincare disk to appear in the mathematical literature. The setting is the Poincare disk model of the hyperbolic plane. We need only a few properties of this model to carry out all of our synthetic constructions: First, the points of the model are the points in the open unit disk in the Euclidean plane. Second, geodesics are circular arcs orthogonal to the unit circle. Inversion across such an arc is to be an isometry. The locus of points equidistant from a given point A is a Euclidean circle (though the Euclidean center of this circle is not A!). This is consistent with our other properties: this locus must be invariant under inversion through any geodesic passing through A; circles orthogonal to every geodesic through A are the only curves satisfying this requirement. Finally, angles in the Poincare model are Euclidean angles. This is the only consistent choice, as inversion preserves Euclidean angles.
