EPINET Reference material

The EPINET project contributes to the enumeration of periodic networks in three–dimensional euclidean space (). 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 () patterns into three–dimensional euclidean space (). The Epinet website is designed to help researchers across many disciplines understand the connections between 2D hyperbolic () and 3D euclidean () 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.

Sodalite on P surface

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 () that we use in all our figures. A nice introduction that requires no mathematics can be found at The Institute for Figuring.

An overview of the EPINET methodology

Mathematical background

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

Related Work

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.
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.