On non-Euclidean crystallography, some football manifolds

Abstract

In polyhedral crystallography, non-Euclidean possibilities come into considerations as well. The so-called football manifolds appearing in the Bolyai–Lobachevskian hyperbolic space (H ³) can model “fullerens” very probably. This may give us the feeling that—in atomic measure—space can wear non-Euclidean structures, e.g., hyperbolic (H ³), spherical (S ³) and other ones, respectively. Such phenomena as pentagonal or decagonal symmetry and the discovery of quasicrystals might strengthen these impressions. We examine here some Archimedean solids: (4, 6, 6) in E ³ (the truncated octahedron in the classical Euclidean space, for analogue as motivation, Fig. 1); (5, 6, 6) the “real football” (now in H ³, Fig. 2, from Molnár in Proceedings of International Conference on Differential Geometry and Its Applications, 1988), (3, 10, 10) in H ³ (Fig. 3 from Molnár in the previous Proceedings); an extra solid (4, 6, 8) with two manifolds on it in H ³ (Figs. 4, 5), published first time in this paper. Furthermore (3, 8, 8) will be realized in the spherical space S ³ (Fig. 6). We can study this classical example as a simple non-Euclidean one also for motivation to our method (see also Molnár in Acta Cryst A61:542–552, 2005, as an introductory paper). Each of them will be a manifold, i.e., it can be endowed with an appropriate face pairing (topological glueing), so that every point has a ball-like neighborhood in the corresponding space, especially of constant curvature, but completely in the classical analogy. In addition, each manifold will be compact (bounded and closed) whose fundamental group can be generated by two screw motions.