Three, four and five-dimensional fullerenes

1999/06/06 by M. Deza, M. I. Shtogrin
Mathematics · #math.CO #math.GT

paper · pdf

published as Southeast Asian Bulletin of Mathematics (1999) 23: 1-10 · 10 pages, 1 figure

arxiv created 1999/06/06 · arxiv updated 2009/11/30

Abstract

We explore some generalizations of fullerenes F_v (simple polyhedra with v vertices and only 5- and 6-gonal faces) seen as (d-1)-dimensional simple manifolds (preferably, spherical or polytopal) with only 5- and 6-gonal 2-faces. First, finite and planar (infinite) 3-fullerenes are described. Three infinite families of spherical 4-fullerenes are presented in Constructions A,B,C. The Construction A gives 4-polytopes by suitable insertion of fullerenes F₃₀(D_5h) into glued 120-cells. The Construction B gives 3-spheres by growing dodecahedra and barrels F₂₄ around of given fullerene. The Construction C gives 4-fullerenes from special decoration of given 4-fullerene, which add facets F₂₀, F₂₄, F₂₆ and F₂₈(T_d) only. Some 5-fullerenes are obtained, by a variation of gluing of two regular tilings 5333 of hyperbolic 4-space or of their suitable quotients.

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