Modelling the relationship between a hyperbolic tessellation and a corresponding triply periodic polyhedron

Abstract

The process of creating artistic patterns has existed for a long time. Many great artists have contributed to this field. Amongst them are notably M.C. Escher, who made many major contributions to this field. He created a few hyperbolic tessellations, which are repeating patterns in hyperbolic geometry and are represented in the Euclidean plane, as formulated by some mathematicians. Creating these patterns was a very complex process, as it had to be done by hand. After his contributions, some programmers developed applications to create these repeating hyperbolic tessellations using the C, C++, and Java programming languages. The motivation for my thesis is to create a simple and interactive tool that allows the user to create these hyperbolic tessellations. The user will have the ability to select the type of hyperbolic tessellation. The user will also be able to create triply periodic polyhedrons in this system. Finally, the system models the relationship between a hyperbolic tessellation and a corresponding triply periodic polyhedron using an interactive and simple mechanism. The tool was created in the Unity Game Engine using the C# programming language.