The football {5, 6, 6} and its geometries: from a sport tool to fullerens and further

TL;DR

This paper explores the geometric properties of football-shaped polyhedra, their symmetry groups, space filling capabilities in Euclidean and hyperbolic spaces, and their relation to Fullerene structures, highlighting open problems in the field.

Contribution

It introduces a novel perspective on football polyhedra, analyzing their space filling properties in hyperbolic space and connecting them to Fullerene structures and symmetry groups.

Findings

Football polyhedra cannot fill Euclidean space E^3.

Hyperbolic space H^3 can be filled with these polyhedra.

Open problems remain in understanding space filling in different geometries.

Abstract

This presentation starts with the regular polygons, of course, then with the Platonic and Archimedean solids. The latter ones are whose symmetry groups are transitive on the vertices, and in addition, whose faces are regular polygons (see only I. Prok's home page [11] for them). Then there come these symmetry groups themselves (starting with the cube and octahedron, of course, then icosahedron and dodecahedron). Then come the space filling properties: Namely the cube is a space filler for the Euclidean space E^3. Then we jump for the other regular solids that cannot fil E^3, but can hyperbolic space H^3, a new space. These can be understood better if we start regular polygons, of course, that cannot fil E^2 in general, but can fil the new plane H2, as hyperbolic or Bolyai-Lobachevsky plane. Now it raises the problem, whether a football polyhedron - with its congruent copies - fil a space. It turns out that E^3 is excluded (it remains an open problem - for you, of course, in other aspects), but H^3 can be filled as a schematic construction show this (Fig. 5), far from elementary. Then we mention some stories on Buckminster Fuller, an architect, who imagined first time fullerens as such crystal structures. Many problems remain open, of course, we are just in the middle of living science.