Descartes Total Angular Defect

The total angular defect is the sum of the Angular Defects over all Vertices of a Polyhedron, where the Angular Defect $\delta$ at a given Vertex is the difference between the sum of face angles and $2\pi$. For any convex Polyhedron, the Descartes total angular defect is

$$\Delta=\sum_i \delta_i = 4\pi.$$ (1)

This is equivalent to the Polyhedral Formula for a closed rectilinear surface, which satisfies

$$\Delta=2\pi(V-E+F).$$ (2)

A Polyhedron with $N_0$ equivalent Vertices is called a Platonic Solid and can be assigned a Schläfli Symbol $\{p, q\}$. It then satisfies

$$N_0={4\pi\over\delta}$$ (3)

and

$$\delta=2\pi-q\left({1-{2\over p}}\right)\pi,$$ (4)

so

$$N_0={4p\over 2p+2q-pq}.$$ (5)

See also Angular Defect, Platonic Solid, Polyhedral Formula, Polyhedron