Density (polytope)

In geometry, the density of a polytope represents the number of windings of a polytope, particularly a uniform or regular polytope, around its center. It can be visually determined by counting the minimum number of facet or face crossings of a ray from the center to infinity. The density is constant across any continuous interior region of a polytope that crosses no facets. For a non-self-intersecting (acoptic) polytope, the density is 1.

Tessellations with overlapping faces can similarly define density as the number of coverings of faces over any given point.

The density of a star polygon is the number of times that the polygonal boundary winds around its center; it is the winding number of the boundary around the central point.

For a regular star polygon {p/q}, the density is q.

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

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 d_v is the density of a vertex figure, d_f of a face and D of the polyhedron as a whole:

For example, the great icosahedron, {3, 5/2}, has 20 triangular faces (d_f = 1), 30 edges and 12 pentagrammic vertex figures (d_v = 2), giving

This implies a density of 7. The unmodified Euler's polyhedron formula fails for the small stellated dodecahedron {5/2, 5} and its dual great dodecahedron {5, 5/2}, for which V − E + F = −6.

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