The Mathematics behind the EPINET Project
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.
The following images show the framework structure of sodalite, which can be embedded on the P surface as shown in the middle. The network forms a tiling of the P surface which unwraps in the hyperbolic plane to a tiling by hexagons.
This project draws on several areas of geometry and crystallography.
The following pages give illustrated and nontechnical explanations of the key concepts.

Mathematical background


A gentle introduction to hyperbolic geometry and the Poincaré
disc model from
The Institute for Figuring.

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.

The technical terms used across this site are explained in the
site
glossary.

Related Work


A brief summary of other approaches to enumerating 3periodic networks, and a note on the history of the EPINET project.

Papers, suggested further reading and links to related websites are collected on the site
bibliography.