Editing Kepler–Poinsot polyhedron (section)

=== Euler characteristic χ ===
A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others.  Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the [[Euler characteristic|Euler relation]]

:<math>\chi=V-E+F=2\ </math>

does not always hold. Schläfli held that all polyhedra must have χ = 2, and he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never widely held.

A modified form of Euler's formula, using [[Density (polytope)|density]] (''D'') of the [[vertex figure]]s (<math>d_v</math>) and faces (<math>d_f</math>) was given by [[Arthur Cayley]], and holds both for convex polyhedra (where the correction factors are all 1), and the Kepler–Poinsot polyhedra:
:<math>d_v V - E + d_f F = 2D.</math>