Gram–Euler theorem
In geometry, the Gram–Euler theorem, Gram-Sommerville, Brianchon-Gram or Gram relation 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 be an -dimensional convex polytope. For each k-face, with its dimension, its interior solid angle is defined by choosing a small enough -sphere centered at some point in the interior of and finding the surface area contained inside. Then the Gram–Euler theorem states: In non-Euclidean geometry of constant curvature the relation gains a volume term, but only if the dimension n is even:Here, is the normalized volume of the polytope ; the angles also have to be expressed as fractions.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:where the first term is the sum of the internal vertex angles, the second sum is over the edges, each of which has internal angle, and the final term corresponds to the entire polygon, which has a full internal angle. For a polygon with faces, the theorem tells us that, or equivalently,. For a polygon on a sphere, the relation gives the spherical surface area or solid angle as the spherical excess:.For a three-dimensional polyhedron the theorem reads:where is the solid angle at a vertex, the dihedral angle at an edge, the third sum counts the faces and the last term is the interior solid angle.