Enumerating uniform polyhedral surfaces with triangular faces Original Research Article

The four infinite sets of planes x+y+z=n, −x+y+z=n, x−y+z=n, x+y+z=n, where n=…−3,−2,−1,0,1,2,3,… divide space into tetrahedral and octahedral regions. A subset of the set of triangular faces of these regions may be chosen so that they form a uniform polyhedral surface, i.e. a surface whose vertices are all equivalent under a group of isometries. There are 26 such surfaces of hyperbolic type; these have 7, 8, 9 or 12 triangles around each vertex.