Euclidean Patterns in NonEuclidean Tilings 
Tilings

Nets

PGD Subgroups: show all  list by orbifold symbol
3D Spacegroups: show all
3D Nets 
The epinet databases currently contain four types of atomic pages: subgroup tilings, Utilings, hnets and snets. The above links connect to lists of these pages organised by symmetry hierarchies. The four types of atomic pages are described below.
Subgroup tilings are hyperbolic tilings defined by their DelaneyDress symbol.
A subgroup tiling page lists the orbifold symbol, vertex, edge, and facetransitivities, vertex degrees, and vertex symbols; it links to its dual tiling page, a single hnet page and pages for each Utiling derived from it.
Utilings are hyperbolic tilings that have the geometry of a specific surfacecompatible symmetry group and translational unit cell. A covering map projects a Utiling onto a periodic minimal surface to define an Etiling. Each (Utiling, covering map) pair defines a distinct Etiling. The edges and vertices of an Etiling define an enet which in turn defines an snet.
A Utiling page contains images of the Utiling and each Etiling and enet derived from it. It links to its dual Utiling, to the snet pages associated with the enets, and to each subgroup tiling that generates the given Utiling.
Hyperbolic hnets are derived from the vertices and edges of a hyperbolic tiling. They have a canonical form given by the maximalsymmetry version of the tiling with the specified vertex degrees and face sizes.
An hnet page lists the maximalsymmetry DelaneyDress symbol, its orbifold symbol, vertex, edge, and facetransitivities, vertex degrees, and vertex symbols; it links to its dual hnet page, to each subgroup tiling that generates the given hnet, and to the snets that arise from this hnet.
Systre nets, or snets are derived from the enets by merging any multigraph edges, computing the equilibrium placement for the periodic net topology, and finding an embedding with maximal symmetry and optimal edgelengths.
An snet page lists its crystallographic spacegroup, unit cell parameters, atom and edge positions, vertex degree and coordination sequence, vertex transitivity, systre key and other names for the structure. It links to each Utiling page that has generated an enet with this topology, and to the hnets that are associated with these Utilings.