## Euclidean Patterns in Non-Euclidean Tilings |

The EPINET project contributes to the enumeration of periodic networks in three–dimensional euclidean space (E³). These networks are of interest to geometers, structural chemists, and statistical physicists.

Instead of working directly in three dimensions, we use the intrinsic hyperbolic geometry of triply periodic minimal surfaces to map two–dimensional hyperbolic (H²) patterns into three–dimensional euclidean space (E³). The Epinet website is designed to help researchers across many disciplines understand the connections between 2D hyperbolic (H²) and 3D euclidean (E³) structure.

As an example, the following images show how the covalent bonding framework of sodalite comes from a tiling on the P surface, which in turn has come from a tiling by hexagons in the hyperbolic plane.

If this is your first encounter with hyperbolic geometry, you might like to make a small detour to learn about the Poincaré disc model of the hyperbolic plane (H²) that we use in all our figures. A nice introduction that requires no mathematics can be found at The Institute for Figuring.

This project draws on several areas of geometry and crystallography. The following pages give illustrated and non-technical explanations of key concepts.

- Minimal surfaces, in general, and those used within EPINET: the primitive, P, diamond, D, and gyroid, G, periodic minimal surfaces.
- The covering maps that wrap the hyperbolic plane onto a triply periodic minimal surface.
- How to enumerate tilings of the hyperbolic plane that are compatible with the covering maps. We use Conway's orbifold notation for 2D discrete symmetry groups, and algorithms from combinatorial tiling theory.
- The surface reticulations that result from projecting the hyperbolic tilings onto each periodic minimal surface and the corresponding aspects of network topology.

- A brief summary of other approaches to enumerating 3-periodic networks, and some historical notes on the EPINET project are given on the related work page.
- Bibliography
- Papers, suggested further reading and links to relevant websites are collected on the site bibliography.
- Glossary of Terms
- The technical terms used across this site are explained in the site glossary.