Four-dimensional football, fullerenes and diagram geometry (Q5943890)

The article of Pasini is an impressive example of how to use methods of diagram geometry for problems in geometry and combinatorics which seem to be far away from diagrams. NEWLINENEWLINENEWLINEGeneralizing the truncated icosahedron (football), the 4-dimensional football is defined to be a tiling \(\Phi\) of a 3-dimensional manifold \(S\) of \(\mathbb{R}^4\) such that (i) the cells of \(\Phi\) are 3-dimensional footballs, (ii) each vertex of \(\Phi\) is contained in four edges, six faces and four cells forming a tetrahedron and (iii) \(S\) is homeomorphic to the sphere of \(\mathbb{R}^4\). NEWLINENEWLINENEWLINETheorem 1. No 4-dimensional footballs exist. NEWLINENEWLINENEWLINETheorem 2. Let \(\Phi\) be a simply connected cell complex with 3-cells and vertices fullfilling conditions (i) and (ii). Then \(\Phi\) is infinite arising as a Grassmann geometry from the Coxeter complex with linear diagram \((5,3,5)\).