Large, Symmetric, “7-Around ” Hyperbolic Disks

Maximal symmetry is used to reduce the computational complexity in the construction of the largest possible hyperbolic disk composed entirely of equilateral triangles meeting seven per vertex. Modeling Hyperbolic Disks Mathematicians as well as artists are fascinated by the hyperbolic plane. Hilbert proved that the infinite hyperbolic plane cannot be smoothly and isometrically embedded (without self-intersections) in 3-space. Artists have made approximate models of a finite portion of the hyperbolic plane from various materials, for instance by crocheting (Fig.1a) or by gluing together paper triangles (Fig.1b). They may start with a small hyperbolic patch (warped like a potato chip) placed at the origin, and then add consecutive annular ribbons around the perimeter. With each annulus the distance from the origin grows only linearly, but the length of the annular ribbon grows exponentially; thus the outer border becomes ever more convoluted, and needs to curl through space in ever tighter undulations. Soon the physical properties of the modeling material will reach some practical limit of packing density, and a uniform radial extension of such a hyperbolic disk is no longer possible. Figure 1: Hyperbolic disks: (a) crochet model by G. Meyer [3], (b) paper model by D. A. Richter [4]