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.

For polygons, density is equal to the sum of exterior angles divided by $2\pi$. The density of a regular polygon {p/q} is q. Examples: {5}(regular pentagon) has the density of $\frac{5\, \cdot \frac{2\pi}{5}}{2\pi} = 1$; {7/3}(regular great heptagram) has the density of $\frac{7\, \cdot \frac{6\pi}{7}}{2\pi} = 3$.

For polyhedra, density is equal to its total curvature (the sum of its angular defects) divided by $4\pi$. Examples: {5,3}(regular dodecahedron) 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}(small stellated dodecahedron) has the density of $\frac{12\cdot\left(2\pi-5\left(\pi-\frac{4\pi}{5}\right)\right)}{4\pi}=3$; {5/2,3}(great stellated dodecahedron) has the density of $\frac{20\cdot\left(2\pi-3\left(\pi-\frac{4\pi}{5}\right)\right)}{4\pi}=7$. If its faces and vertex figures are non-self-intersecting, the density is also half the Euler characteristic.