Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture (2016)

Pages 263–268

Euler’s formula for a polyhedron with *V* vertices, *F* faces and *E* edges and genus *g* states that *V* + *F* − *E* = 2(1 − *g*).
Cayley formulated a similar expression for the four Kepler-Poinsot polyhedra:
*bV* + *aF* − *E* = 2*c*, using the
densities
*a*, *b* and *c* of respectively the faces, the vertices and the polyhedron itself.
We conjecture both should be united as
*bV* + *aF* − *E* = 2*c*(1 &minus *g*),
or, more generally, for Archimedean polyhedra with *V _{j}*
vertices and

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