**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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