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.
Statement[edit]
Let be an -dimensional convex polytope. For each k-face , with 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,
, and
hyperbolic,
, geometry) the relation gains a volume term, but only if the dimension
n is even:
Here,
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]
Examples[edit]
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 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
).
History[edit]
The n-dimensional relation was first proven by Sommerville, Heckman and Grünbaum for the spherical, hyperbolic and Euclidean case, respectively.[2]
See also[edit]
References[edit]