## 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 at a given Vertex is the difference between the sum of face angles and . For any convex Polyhedron, the Descartes total angular defect is

 (1)

This is equivalent to the Polyhedral Formula for a closed rectilinear surface, which satisfies
 (2)

A Polyhedron with equivalent Vertices is called a Platonic Solid and can be assigned a Schläfli Symbol . It then satisfies

 (3)

and
 (4)

so
 (5)

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

© 1996-9 Eric W. Weisstein
1999-05-24