Euler’s polyhedron formula for tessellations
Dirk Huylebrouck

Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture
Pages 211–216
Regular Papers

Abstract

In 1758 Euler stated his formula for a polyhedron on a sphere with V vertices, F faces and E edges: V + FE = 2. Since then, it has been extended to many more cases, for instance, to connected plane graphs by including the ‘exterior face’ in the number of faces, or to ‘infinite polyhedra’, using their genus g. For g = 1 the latter generalization becomes V + FE = 0 or V + F = E and that is the case for uniform planar tessellations. This way, two popular topics, uniform planar tessellations and Euler’s formula, can be combined.

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