Density

Not to be confused with Dense polytope.

The density of an orientable polytope is an integer that generalizes the concept of a turning number of a polygon. Density is well-defined for orientable uniform polytopes with no faces passing through the center (as in the hemipolyhedra), but standardization of density is poor for general polytopes, especially those without a well-defined center.

Polygon density

For polygons, density is equal to the sum of exterior angles divided by $2\pi$. The density of a regular polygon $\{p/q\}$, where $p/q$ is an irreducible fraction, is q .

Examples

• {5} has the density of ${\frac {5\,\cdot {\frac {2\pi }{5}}}{2\pi }}=1$
• {7/3} has the density of ${\frac {7\,\cdot {\frac {6\pi }{7}}}{2\pi }}=3$.

Polyhedron density

For polyhedra, density is equal to its total curvature (the sum of its angular defects) divided by $4\pi$.

If the faces and vertex figures of a polyhedron are non-self-intersecting, the density is also half the Euler characteristic.

Examples

• {5,3} has the density of ${\frac {20\cdot \left(2\pi -3\left(\pi -{\frac {2\pi }{5}}\right)\right)}{4\pi }}=1$
• {5/2,5} has the density of ${\frac {12\cdot \left(2\pi -5\left(\pi -{\frac {4\pi }{5}}\right)\right)}{4\pi }}=3$
• {5,5/2} has the density of ${\frac {12\cdot \left(4\pi -5\left(\pi -{\frac {2\pi }{5}}\right)\right)}{4\pi }}=3$
• {5/2,3} has the density of ${\frac {20\cdot \left(2\pi -3\left(\pi -{\frac {4\pi }{5}}\right)\right)}{4\pi }}=7$.