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 {\displaystyle P} be an n {\displaystyle n} -dimensional convex polytope. For each k-face F {\displaystyle F} , with k = dim ( F ) {\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 ∠ ( F ) {\displaystyle \angle (F)} is defined by choosing a small enough ( n − 1 ) {\displaystyle (n-1)} -sphere centered at some point in the interior of F {\displaystyle F} and finding the surface area contained inside P {\displaystyle P} . Then the Gram–Euler theorem states: ∑ F ⊂ P ( − 1 ) dim F ∠ ( F ) = 0 {\displaystyle \sum _{F\subset P}(-1)^{\dim F}\angle (F)=0} In non-Euclidean geometry of constant curvature (i.e. spherical, ϵ = 1 {\displaystyle \epsilon =1} , and hyperbolic, ϵ = − 1 {\displaystyle \epsilon =-1} , geometry) the relation gains a volume term, but only if the dimension n is even: ∑ F ⊂ P ( − 1 ) dim F ∠ ( F ) = ϵ n / 2 ( 1 + ( − 1 ) n ) Vol ( P ) {\displaystyle \sum _{F\subset P}(-1)^{\dim F}\angle (F)=\epsilon ^{n/2}(1+(-1)^{n})\operatorname {Vol} (P)} Here, Vol ( P ) {\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 ∠ ( F ) {\displaystyle \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: ∑ v α v − ∑ e π + 2 π = 0 {\displaystyle \sum _{v}\alpha _{v}-\sum _{e}\pi +2\pi =0} where the first term A = ∑ α v {\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 π {\displaystyle \pi } , and the final term corresponds to the entire polygon, which has a full internal angle 2 π {\displaystyle 2\pi } . For a polygon with n {\displaystyle n} faces, the theorem tells us that A − π n + 2 π = 0 {\displaystyle A-\pi n+2\pi =0} , or equivalently, A = π ( n − 2 ) {\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: Ω = A − π ( n − 2 ) {\displaystyle \Omega =A-\pi (n-2)} . For a three-dimensional polyhedron the theorem reads: ∑ v Ω v − 2 ∑ e θ e + ∑ f 2 π − 4 π = 0 {\displaystyle \sum _{v}\Omega _{v}-2\sum _{e}\theta _{e}+\sum _{f}2\pi -4\pi =0} where Ω v {\displaystyle \Omega _{v}} is the solid angle at a vertex, θ e {\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 2 π {\displaystyle 2\pi } ) and the last term is the interior solid angle (full sphere or 4 π {\displaystyle 4\pi } ).
History The n-dimensional relation was first proven by Sommerville, Heckman and Grünbaum for the spherical, hyperbolic and Euclidean case, respectively.
See also Euler characteristic Dehn–Sommerville equations Angular defect Gauss–Bonnet theorem
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