Equivelar Polyhedra with Few Vertices

Abstract

We know that the polyhedra corresponding to the Platonic solids are equivelar. In this article we have classified completely all the simplicial equivelar polyhedra on ≤ 11 vertices. There are exactly 27 such polyhedra. For each n ≥ −4, we have classified all the ( p, q) such that there exists an equivelar polyhedron of type { p, q} and of Euler characteristic n. We have also constructed five types of equivelar polyhedra of Euler characteristic −2m, for each m ≥ 2.

Degree-regular triangulations of the double-torus

“…Now, {2, 3, 4} is a face of N 1 but ϕ({2, 3, 4}) = {1, 2, 6} is not a face of N 3 , a contradiction. If ϕ(1) = 9 then from lk N 1 (0), lk N 1 (1), lk N 3 (0) and lk N 3 (9), we get ϕ = (1, 9, 5, 3, 7)(2, 10) (4,8). Now, {5, 7, 11} is a face of N 1 but ϕ({5, 7, 11}) = {1, 3, 11} is not a face of N 3 , a contradiction.…”

“…, where α = (0, 4, 8)(1, 5, 9)(2, 6, 10) (3,7,11), β = (0, 2)(4, 10)(1, 11)(3, 9)(5, 7) (6,8), α 1 = (1, 2)(3, 4)(5, 6)(7, 8)(9, 10)(11, 0), α 2 = (1, 3)(2, 4)(5, 7)(6, 8)(9, 11)(10, 0) and γ = (1, 2)(3, 4)(5, 6)(7, 8)(9, 10).…”

“…In [3], Bokowski and Guedes de Oliveira have shown that one of these 59 combinatorial 2-manifolds (namely, N 12 54 ) is not geometrically realizable in R 3 . In [8], Datta and Nilakantan have shown that there are exactly 27 degree-regular combinatorial 2-manifolds on at most 11 vertices.…”

“…So, if K is weakly regular and χ(K) ≥ 0 then (n, d) = (4,3), (6,4), (6,5), (12,5) (6,4), (6,5), (12,5)}, there exists a unique combinatorial 2-manifold, namely, the 4-vertex 2-sphere, the boundary of the octahedron, the 6-vertex real projective plane and the boundary of the icosahedron (cf. [5,8]). These 4 combinatorial 2-manifolds are combinatorially regular.…”

“…There exists an n-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9 and for k ≥ 2, there exists a (4k + 2)-vertex weakly regular triangulation of the Klein bottle (cf. [8,9]). There are infinitely many combinatorially regular triangulations of the torus whereas there is no such triangulation of the Klein bottle (cf.…”