About 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.
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.
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.
-
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.
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.
- References
-
Papers and links to relevant websites are collected on the site
references page.
- Glossary of Terms
-
The technical terms used across this site are explained in the
site
glossary.
The EPINET website structure
The EPINET website is based on two databases:
The all-important links between 2D hyperbolic and 3D euclidean structure are recorded in two sets of relations between these databases:
- A map from each hyperbolic surface reticulation to its 3D network topology. For surface reticulations with neighbour collisions this map is empty, because it is not possible to find a systre canonical form for the network structure.
- The inverse map from each 3D network topology to all the surface reticulations that have the given topology.
EPINET implements a number of different ways to explore these databases and the relations between them.
- 2D hyperbolic tiling index
-
The tilings index is a listing of all the H² tilings currently stored within EPINET.
Each hyperbolic tiling page has links to the corresponding surface tilings, and the resulting 3D network topology-type. Note that it is possible to switch between tile- and vertex-transitive versions of each hyperbolic tiling and surface reticulation.
-
- 3D network index
-
The networks index is a listing of every collision-free network topology found via the hyperbolic surface tiling approach. Each 3D network page lists all the surface reticulations that have the given topology.
- PGD Subgroups index
-
The PGD subgroups index is a listing of the subgroups that are compatible with the genus-3 translational unit of the the P, D, and G, surfaces. The subgroups are listed by their orbifold symbol. On each subgroup page there is a list of links to each
1- and 2-transitive tiling with that symmetry.
- Search pages
-
From the 2D search and
3D search pages it is possible to initiate queries on either the 2D hyperbolic tiling or 3D network databases.
- Known 3D networks
-
The known networks page lists some of the networks represented in EPINET that are already well known to crystallographers. The list is somewhat random and incomplete, and we welcome any new additions that you might find —
just contact us.
From a familiar net, such as the simple cubic lattice, you can explore related structures obtained by wrapping the same hyperbolic tiling onto a different surface, for example.
Technical aspects
The databases are maintained using MySQL and the EPINET website uses Ruby-on-Rails to generate the html pages.
Revisions to the website content are posted on the
What's new? page.
The people involved
Many people have contributed in essential ways, we thank them on the
acknowledgements page.
If you have questions or suggestions about the EPINET project, please
contact us.
