Three, four and five-dimensional fullerenes

TL;DR

This paper generalizes the concept of fullerenes to higher dimensions, describing new families of 3-, 4-, and 5-dimensional structures with specific face properties, and introduces methods for constructing these complex polyhedral forms.

Contribution

It introduces novel constructions for higher-dimensional fullerenes, expanding the understanding of their geometric and topological properties beyond traditional 3D cases.

Findings

Presented three infinite families of spherical 4-fullerenes.

Constructed 4-polytopes by inserting fullerenes into 120-cells.

Generated 3-spheres and 4-fullerenes through specific decoration and gluing methods.

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 F_{30}(D_{5h}) into glued 120-cells. The Construction B gives 3-spheres by growing dodecahedra and barrels F_{24} around of given fullerene. The Construction C gives 4-fullerenes from special decoration of given 4-fullerene, which add facets F_{20}, F_{24}, F_{26} and F_{28}(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.