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

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, polytope density represents the number of windings

Degree of a continuous mapping

In topology, the degree is a numerical invariant that describes a continuous mapping between two compact oriented manifolds of the same dimension. Intuitively, the degree represents the number of times that the domain manifold wraps around the range manifold under the mapping...

 of a polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

, particularly a uniform

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

 or regular polytope

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

, 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.

Polyhedra

Arthur Cayley

Arthur Cayley

Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

 used density as a way to correct the Polyhedron formula for nonconvex uniform polyhedron

Nonconvex uniform polyhedron

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...

s, where d_v is the density of the vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 and d_f is the density of the face and D is the density of the polyhedron.

d_v * V + d_f * F = E + 2 * D

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

2*12+1*20 = 44 = 30 + 2 * D.

This implies a density of 7.