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Descartes Total Angular Defect


The total angular defect is the sum of the angular defects over all polyhedron vertices of a polyhedron, where the angular defect delta at a given polyhedron vertex is the difference between the sum of face angles and 2pi. For any convex polyhedron, the Descartes total angular defect is

 Delta=sum_(i)delta_i=4pi.
(1)

This is equivalent to the polyhedral formula for a closed rectilinear surface, which satisfies

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

A polyhedron with N_0 equivalent polyhedron vertices is called a Platonic solid and can be assigned a Schläfli symbol {p,q}. It then satisfies

 N_0=(4pi)/delta
(3)

and

 delta=2pi-q(1-2/p)pi,
(4)

so

 N_0=(4p)/(2p+2q-pq).
(5)

See also

Angular Defect, Platonic Solid, Polyhedral Formula, Polyhedron

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Cite this as:

Weisstein, Eric W. "Descartes Total Angular Defect." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DescartesTotalAngularDefect.html

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