We first study properties of the regular net which are important for the construction of certain triangulations of the 2-sphere (subdivisions of the icosahedron). For a triangulation of the 2-sphere we attach to each vertex the "multiplicity" 6-r where r is the number of triangles meeting in the vertex. By Euler's polyhedral theorem the sum of all multiplicities is always 12.
Belonging to a triangulation there is a metric which makes each triangle flat and equilateral. The "multiplicity" 6-r of a vertex corresponds to the curvature concentrated in the vertex. If r <= 6 for each vertex then we have non-negative curvature. William P. Thurston has studied systematically triangulations with non-negative curvature. I shall try to give some idea of this work. The dual cell decomposition of a triangulation has the property that in each vertex three faces come together. Each vertex of the triangulation (with r triangles meeting) produces an r-gon as a face of the dual decomposition. If r takes only the values 5 and 6, we obtain the fullerenes used in carbon chemistry. Special fullerenes will be studied. Triangulations or their duals are also used to describe biological structures.Friedrich Hirzebruch