Gram–Euler theorem

In geometry, the Gram–Euler theorem, Gram-Sommerville, Brianchon-Gram or Gram relation (named after Jørgen Pedersen Gram, Leonhard Euler, Duncan Sommerville and Charles Julien Brianchon) is a generalization of the internal angle sum formula of polygons to higher-dimensional polytopes. The equation constrains the sums of the interior angles of a polytope in a manner analogous to the Euler relation on the number of d-dimensional faces.

Statement
Let $$P$$ be an $$n$$-dimensional convex polytope. For each k-face $$F$$, with $$k = \dim(F)$$ its dimension (0 for vertices, 1 for edges, 2 for faces, etc., up to n for P itself), its interior (higher-dimensional) solid angle $$\angle(F)$$ is defined by choosing a small enough $$(n - 1)$$-sphere centered at some point in the interior of $$F$$ and finding the surface area contained inside $$P$$. Then the Gram–Euler theorem states: $$\sum_{F \subset P} (-1)^{\dim F} \angle(F) = 0$$In non-Euclidean geometry of constant curvature (i.e. spherical, $$\epsilon = 1$$, and hyperbolic, $$\epsilon = -1$$, geometry) the relation gains a volume term, but only if the dimension n is even:$$\sum_{F \subset P} (-1)^{\dim F} \angle(F) = \epsilon^{n/2}(1 + (-1)^n)\operatorname{Vol}(P)$$Here, $$\operatorname{Vol}(P)$$ is the normalized (hyper)volume of the polytope (i.e, the fraction of the n-dimensional spherical or hyperbolic space); the angles $$\angle(F)$$ also have to be expressed as fractions (of the (n-1)-sphere).

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

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

For a three-dimensional polyhedron the theorem reads:$$\sum_{v} \Omega_v - 2\sum_e \theta_e + \sum_f 2\pi - 4\pi = 0$$where $$\Omega_v$$ is the solid angle at a vertex, $$\theta_e$$ 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 $$2\pi$$) and the last term is the interior solid angle (full sphere or $$4\pi$$).

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