Establishing the Connection Between a Triply Periodic Polyhedron and its Hyperbolic Covering Tessellation

Abstract

The physical world we live in is usually portrayed using the five postulates of Euclidean geometry. Endeavors to infer the fifth postulate from other four led to the discovery of hy- perbolic geometry in 19th century. However some facts about hyperbolic geometry, for example the sum of angles in a traingle is less than 180 degrees in hyperbolic geometry, contradict our Euclidean knowledge. These unusual properties in turn intrigued many math- ematicians to carry out research on its hyperbolic patterns as they are hard to visualize. Patterns called hyperbolic tessellations are the basis for this research. Research so far on hyperbolic tessellations has led to the development of many applications using various plat- forms and programming languages, but all the applications generate tessellations based on many input parameters and are difficult for a user who does not have much insight into hy- perbolic geomery. This thesis concentrates on developing an application which simplifies this process and as an enhancement helps users to view the three dimensional model of the hyperbolic tessellation that is generated. The application developed as part of this thesis provides visualization in two sub-screens. The first one contains a hyperbolic tessellation with a selected motif on it, and the second screen contains a three dimensional triply periodic polyhedral model of the hyperbolic tes- sellation. Users need to specify number of sides of a polygon and number of polygons that meet at each vertex in hyperbolic tessellation. Based on the input, the application generates layers of the hyperbolic tessellation in incremental order in a two-dimensional space and provides a three-dimensional triply periodic polyhedral representation. The visulization was implemented using Unity Pro game engine and C# as the developing language.