Enumerating uniform polyhedral surfaces with triangular faces (Q1842171)

This very nice paper describes a family of embedded polyhedral surfaces in \(E^ 3\) consisting of equilateral triangles. Each surface is uniform in the sense that the Euclidean group (including reflections) acts transitively on the set of vertices. The faces lie in four families of parallel planes, each vertex figure is part of a cuboctahedron. By enumerating the possible simple closed paths in the cuboctahedron, the author obtains an enumeration of 26 uniform polyhedral surfaces with 7, 8, 9 or 12 triangles around each vertex. Combinatorially, these are nothing but the regular hyperbolic tessellations \(\{3,7\}\), \(\{3,8\}\), \(\{3,9\}\) or \(\{3,12\}\), respectively. Finite uniform polyhedral surfaces with self-intersections as well as infinite ones without self- intersections (consisting of squares or hexagonal) were described in various papers by Coxeter as so-called regular skew polyhedra [compare \textit{H. S. M. Coxeter}, Proc. Lond. Math. Soc., II. Ser. 43, 33-62 (1937; Zbl 0016.27101)].