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This presentation starts with the regular polygons, of course, then with thePlatonic and Archimedean solids. The latter ones are whose symmetry groups aretransitive 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 symmetrygroups themselves (starting with the cube and octahedron, of course, thenicosahedron and dodecahedron). Then come the space filling properties: Namelythe cube is a space filler for the Euclidean space E^3. Then we jump for theother regular solids that cannot fil E^3, but can hyperbolic space H^3, a newspace. 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 orBolyai-Lobachevsky plane. Now it raises the problem, whether a footballpolyhedron - with its congruent copies - fil a space. It turns out that E^3 isexcluded (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 fromelementary. Then we mention some stories on Buckminster Fuller, an architect,who imagined first time fullerens as such crystal structures. Many problemsremain open, of course, we are just in the middle of living science.

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