| The nonconvex great icosahedron, {3,5/2} has a density of 7 as demonstrated in this transparent and cross-sectional view on the right. | |
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 crossings of a ray from the center to infinity. For non-intersecting polytopes, the density is 1.
For hemipolyhedra, which pass through the center, the density is not well-defined, though the center may be counted as half a point, or more precisely one defines the density as half the number of facet crossings of a line from infinity, through the center, passing out the other side to infinity.
Each continuous interior region of a polytope that crosses no facets can be seen as an integer density of one or higher.
Tessellations with overlapping faces can similarly define density as the number of coverings of faces over any given point.
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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 correct the Polyhedron formula for nonconvex uniform polyhedrons, where dv is the density of the vertex figure and df is the density of the face and D is the density of the polyhedron.
For example, the great icosahedron, {3,5/2}, has 20 triangular faces (df=1), and 12 pentagrammic vertex figures (dv=2):
This implies a density of 7.
There are 4 regular star polyhedra (called the Kepler–Poinsot solids), which have a densities of 4 or 7.
There are 10 regular star polychora or 4-polytopes (called the Schläfli–Hess polychora), which have densities between 4 and 191.