Density (polytope)

In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets. The same calculation can be performed for any convex polyhedron, even one without symmetries, by choosing any point interior to the polyhedron as its center. For these polyhedra, the density will be 1. More generally, for any non-self-intersecting (acoptic) polyhedron, the density can be computed as 1 by a similar calculation that chooses a ray from an interior point that only passes through facets of the polyhedron, adds one when this ray passes from the interior to the exterior of the polyhedron, and subtracts one when this ray passes from the exterior to the interior of the polyhedron. 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. Tessellations with overlapping faces can similarly define density as the number of coverings of faces over any given point. The density of a polygon is the number of times that the polygonal boundary winds around its center. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem. The density of a polygon can also be called its turning number; the sum of the turn angles of all the vertices divided by 360°. This will be an integer for all unicursal paths in a plane. The density of a compound polygon is the sum of the densities of the component polygons. For a regular star polygon {p/q}, the density is q.