Gram–Euler theorem

In geometry, the Gram–Euler theorem generalizes the internal angle sum formula to higher-dimensional polytopes.

Statement

Let ${\displaystyle P}$ be an ${\displaystyle n}$-dimensional convex polytope. For each cell ${\displaystyle E}$, let ${\displaystyle \dim(E)}$ be its dimension (0 for vertices, 1 for edges, 2 for faces, etc.), and ${\displaystyle \angle (E)}$ be its internal solid angle, determined by choosing a small enough ${\displaystyle (n-1)}$-sphere centered at some point in the interior of ${\displaystyle E}$ and finding the surface area contained inside ${\displaystyle P}$. Then, ${\textstyle \sum _{E\in \operatorname {Cells} (P)}(-1)^{\dim(E)}\angle (E)=0}$.[1]

Examples

For a polygon ${\displaystyle P}$ with ${\displaystyle n}$ sides, there is one face (the entire polygon), which has internal angle ${\displaystyle 2\pi }$, and ${\displaystyle n}$ edges, each of which has internal angle ${\displaystyle \pi }$. Let ${\displaystyle A}$ be the sum of the internal angles of the corners. The Gram-Euler theorem then tells us that ${\displaystyle A-\pi n+2\pi =0}$, or equivalently, ${\displaystyle A=\pi (n-2)}$.

References

1. Grünbaum, Branko. Convex Polytopes. Springer. pp. 297–303. ISBN 978-0-387-40409-7.