The regular net of triangles in the plane and applications, for example to carbon molecules

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

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