Convex and discrete geometry Polytopes and polyhedra Theorem

Descartes's theorem on total angular defect

Builds on Convex polytope Vertex of a polytope Euler characteristic

Statement

Let $P \subseteq \R^3$ be a three-dimensional convex polytope. Let $v$ be a vertex of $P$ . For a face $F$ of $P$ that contains $v$ , let $\alpha_F(v)$ be the interior angle of the polygon $F$ at $v$ . The angular defect at $v$ is $\delta(v) = 2\pi - \sum_F \alpha_F(v),$ where the sum runs over the faces $F$ of $P$ that contain $v$ . Then $\sum_{v} \delta(v) = 4\pi,$ where the sum runs over all vertices of $P$ .

Remarks

Let $P$ have $V$ vertices, $E$ edges and $F$ faces. A face with $k$ sides has angle sum $(k - 2)\pi$ , and the numbers of sides of all faces add up to $2E$ . So all face angles of $P$ add up to $2\pi E - 2\pi F$ . Hence $\sum_v \delta(v) = 2\pi V - 2\pi E + 2\pi F = 2\pi (V - E + F)$ , and Euler's polyhedron formula $V - E + F = 2$ gives $4\pi$ . The same count shows that for a polyhedral surface of Euler characteristic $\chi$ the total defect is $2\pi\chi$ .

Example: the cube has $8$ vertices, each with three right angles, so each defect is $2\pi - 3\pi/2 = \pi/2$ , and the total is $8 \cdot \pi/2 = 4\pi$ . The regular tetrahedron has $4$ vertices with three angles of $\pi/3$ each, so each defect is $\pi$ , and the total is again $4\pi$ .

Convexity enters only through Euler's formula: the statement holds for every polyhedron whose surface is a topological sphere, where a defect may be negative at a non-convex vertex.

Going deeper

For a student

The theorem is the discrete form of the Gauss–Bonnet theorem for the sphere: the defect at a vertex plays the role of the total Gaussian curvature concentrated there. It is equivalent to Euler's formula, given the angle count in the proof. It shows that a convex polyhedron has at least one vertex with defect at least $4\pi/V$ , and it constrains the possible vertex configurations of Platonic and Archimedean solids. It also explains why the total of the cone angles is fixed in Alexandrov's theorem on polyhedra.

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Convex polytope concept
Euler characteristic concept
Vertex of a polytope concept
Descartes's theorem on total angular defect statement

Tag	`PT67C5ANSD`
URI	`m2://PT67C5ANSD`
Kind	statement, theorem
Writing	W0 stub
MSC	52B10, 51M20
Status	This copy of the site has no build data. Run `m2 status PT67C5ANSD`.
About	Convex polytope
Uses	Convex polytope, Vertex of a polytope, Euler characteristic