Density (polytope)

Content sourced from Wikipedia, licensed under CC BY-SA 3.0.

Density (polytope) extends the idea of winding around a center from circles to steadier shapes in higher dimensions. It tells you how many times a polyhedron wraps around its center.

How to measure it simply: pick a point at the center and shoot a ray to infinity. Count how many facets (faces of the appropriate dimension) the ray crosses, making sure the ray only passes through facets and not through lower‑dimensional features. If this crossing count is the same no matter which ray you choose and the center isn’t sitting on a facet, that count is the density. For ordinary convex polyhedra this method gives density 1. The same idea works for any convex body inside, even without symmetry, by choosing an interior point as center.

For non-self-intersecting (acoptic) polyhedra a similar ray method can be used and also yields density 1. But star polyhedra are trickier because they don’t have a clear interior and exterior.

Density can also be defined for polygons: it is how many times the boundary winds around the center. For convex or simple polygons this density is 1. It’s related to the turning number, which is the total amount the direction of the boundary turns, divided by 360 degrees. A regular star polygon {p/q} has density q. If a polygon is a compound made of several pieces, its density is the sum of the densities of the pieces.

Density is the same for a polyhedron and its dual.

There is a strong link between density and curvature. A polyhedron can be viewed as a surface where curvature concentrates at the vertices (angle defects). The density equals the total curvature divided by 4π. For example, a cube has eight vertices with a defect of π/2 each, giving a total of 4π and density 1.

For simple-face, simple-vertex polyhedra, density equals half the Euler characteristic. If the surface has genus g, its density is 1 − g.

Density also helps extend Euler’s formula to star polyhedra (where the usual V − E + F = 2 fails). Cayley proposed a way to modify the formula using densities of vertex figures and faces, which explains why some star polyhedra have densities like 3 or 7. This approach also shows why two duals can share the same density, though the unmodified Euler rule breaks down for some stars.

In the broader family of uniform polyhedra, Coxeter and others used density to classify many shapes, although some types (like hemipolyhedra, whose faces pass through the center, or non-orientable polyhedra) do not have a well-defined density.

There are 10 regular star 4-polytopes (the Schläfli–Hess 4-polytopes). Their densities vary widely, and they come in dual pairs, with two self-dual exceptions having densities 6 and 66.