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.