In geometry, the Gram–Euler theorem,[1] Gram-Sommerville, Brianchon-Gram or Gram relation[2] (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.
-dimensional convex polytope.
its dimension (0 for vertices, 1 for edges, 2 for faces, etc., up to n for P itself), its interior (higher-dimensional) 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:[3][1]
In non-Euclidean geometry of constant curvature (i.e. spherical,
, geometry) the relation gains a volume term, but only if the dimension n is even:
is the normalized (hyper)volume of the polytope (i.e, the fraction of the n-dimensional spherical or hyperbolic space); the angles
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:
π + 2 π = 0
is the sum of the internal vertex angles, the second sum is over the edges, each of which has internal angle
π
, and the final term corresponds to the entire polygon, which has a full internal angle
2 π
faces, the theorem tells us that
− π n + 2 π = 0
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:
is the solid angle at a vertex,
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
) and the last term is the interior solid angle (full sphere or
The n-dimensional relation was first proven by Sommerville, Heckman and Grünbaum for the spherical, hyperbolic and Euclidean case, respectively.