Gram–Euler theorem

Let ${\displaystyle P}$ be an ${\displaystyle n}$-dimensional convex polytope. For each k-face ${\displaystyle F}$, with ${\displaystyle k=\dim(F)}$ its dimension (0 for vertices, 1 for edges, 2 for faces, etc., up to n for P itself), its interior (higher-dimensional) solid angle ${\displaystyle \angle (F)}$ is defined by choosing a small enough ${\displaystyle (n-1)}$-sphere centered at some point in the interior of ${\displaystyle F}$ and finding the surface area contained inside ${\displaystyle P}$. Then the Gram–Euler theorem states:[3][1]

$${\displaystyle \sum _{F\subset P}(-1)^{\dim F}\angle (F)=0}$$

In non-Euclidean geometry of constant curvature (i.e. spherical, ${\displaystyle \epsilon =1}$, and hyperbolic, ${\displaystyle \epsilon =-1}$, geometry) the relation gains a volume term, but only if the dimension n is even:

$${\displaystyle \sum _{F\subset P}(-1)^{\dim F}\angle (F)=\epsilon ^{n/2}(1+(-1)^{n})\operatorname {Vol} (P)}$$

Here, ${\displaystyle \operatorname {Vol} (P)}$ is the normalized (hyper)volume of the polytope (i.e, the fraction of the n-dimensional spherical or hyperbolic space); the angles ${\displaystyle \angle (F)}$ also have to be expressed as fractions (of the (n-1)-sphere).[2]

When the polytope is simplicial additional angle restrictions known as Perles relations hold, analogous to the Dehn-Sommerville equations for the number of faces.[2]

For a two-dimensional polygon, the statement expands into:

$${\displaystyle \sum _{v}\alpha _{v}-\sum _{e}\pi +2\pi =0}$$

where the first term ${\displaystyle A=\textstyle \sum \alpha _{v}}$ is the sum of the internal vertex angles, the second sum is over the edges, each of which has internal angle ${\displaystyle \pi }$, and the final term corresponds to the entire polygon, which has a full internal angle ${\displaystyle 2\pi }$. For a polygon with ${\displaystyle n}$ faces, the theorem tells us that ${\displaystyle A-\pi n+2\pi =0}$, or equivalently, ${\displaystyle A=\pi (n-2)}$. For a polygon on a sphere, the relation gives the spherical surface area or solid angle as the spherical excess: ${\displaystyle \Omega =A-\pi (n-2)}$. For a three-dimensional polyhedron the theorem reads:

$${\displaystyle \sum _{v}\Omega _{v}-2\sum _{e}\theta _{e}+\sum _{f}2\pi -4\pi =0}$$

where ${\displaystyle \Omega _{v}}$ is the solid angle at a vertex, ${\displaystyle \theta _{e}}$ the dihedral angle at an edge (the solid angle of the corresponding lune is twice as big), the third sum counts the faces (each with an interior hemisphere angle of ${\displaystyle 2\pi }$) and the last term is the interior solid angle (full sphere or ${\displaystyle 4\pi }$).

The n-dimensional relation was first proven by Sommerville, Heckman and Grünbaum for the spherical, hyperbolic and Euclidean case, respectively.[2]

• Euler characteristic
• Dehn-Sommerville equations
• Angular defect
• Gauss-Bonnet theorem