# Density

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 ${\displaystyle 2\pi }$. The density of a regular polygon ${\displaystyle \{p/q\}}$, where ${\displaystyle p/q}$ is an irreducible fraction, is q .

### Examples

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

## Polyhedron density

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

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