Density (polytope)

However, this assignment of signs to crossings does not generally apply to star polyhedra, as they do not have a well-defined interior and exterior.

It can be visually determined by counting the minimum number of edge crossings of a ray from the center to infinity.

A polyhedron can be considered a surface with Gaussian curvature concentrated at the vertices and defined by an angle defect.

The density of a polyhedron is equal to the total curvature (summed over all its vertices) divided by 4π.

[2] For example, a cube has 8 vertices, each with 3 squares, leaving an angle defect of π/2.

The density of a polyhedron with simple faces and vertex figures is half of the Euler Characteristic, χ.

If its genus is g, its density is 1-g. Arthur Cayley used density as a way to modify Euler's polyhedron formula (V − E + F = 2) to work for the regular star polyhedra, where dv is the density of a vertex figure, df of a face and D of the polyhedron as a whole: For example, the great icosahedron, {3, 5/2}, has 20 triangular faces (df = 1), 30 edges and 12 pentagrammic vertex figures (dv = 2), giving This implies a density of 7.

Edmund Hess generalized the formula for star polyhedra with different kinds of face, some of which may fold backwards over others.

The resulting value for density corresponds to the number of times the associated spherical polyhedron covers the sphere.