# Density

For polygons, density is equal to the sum of exterior angles divided by ${\displaystyle 2\pi}$. The density of a regular polygon {p/q} is q. Examples: {5}(regular pentagon) has the density of ${\displaystyle \frac{5\, \cdot \frac{2\pi}{5}}{2\pi} = 1}$; {7/3}(regular great heptagram) has the density of ${\displaystyle \frac{7\, \cdot \frac{6\pi}{7}}{2\pi} = 3}$.
For polyhedra, density is equal to its total curvature (the sum of its angular defects) divided by ${\displaystyle 4\pi}$. Examples: {5,3}(regular dodecahedron) 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}(small stellated dodecahedron) has the density of ${\displaystyle \frac{12\cdot\left(2\pi-5\left(\pi-\frac{4\pi}{5}\right)\right)}{4\pi}=3}$; {5/2,3}(great stellated dodecahedron) has the density of ${\displaystyle \frac{20\cdot\left(2\pi-3\left(\pi-\frac{4\pi}{5}\right)\right)}{4\pi}=7}$. If its faces and vertex figures are non-self-intersecting, the density is also half the Euler characteristic.