Generalization of Euler’s formula for adjacent polyhedra

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Abstract
Euler's formula for polyhedra is one of the most famous mathematical results. It is also widely used outside mathematics. In particular, it is used in the analysis of molecular structures, many of which have a polyhedral shape. Generalizations of Euler's formula for non-simply connected and multidimensional polyhedra are well known. This article presents a generalization of Euler's formula for adjacent polyhedra. Various cases of adjacency are considered: face-sharing, edge-sharing and vertex-sharing connections. For a system of adjacent polyhedra, a single formula relates the number of vertices, edges and faces in the form V – E + F = N + 1, where N is the number of polyhedra.

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