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 [math]\displaystyle{ P }[/math] be an [math]\displaystyle{ n }[/math]-dimensional convex polytope. For each k-face [math]\displaystyle{ F }[/math], with [math]\displaystyle{ k = \dim(F) }[/math] its dimension (0 for vertices, 1 for edges, 2 for faces, etc., up to n for P itself), its interior (higher-dimensional) solid angle [math]\displaystyle{ \angle(F) }[/math] is defined by choosing a small enough [math]\displaystyle{ (n - 1) }[/math]-sphere centered at some point in the interior of [math]\displaystyle{ F }[/math] and finding the surface area contained inside [math]\displaystyle{ P }[/math]. Then the Gram–Euler theorem states: [math]\displaystyle{ \sum_{F \subset P} (-1)^{\dim F} \angle(F) = 0 }[/math]In non-Euclidean geometry of constant curvature (i.e. spherical, [math]\displaystyle{ \epsilon = 1 }[/math], and hyperbolic, [math]\displaystyle{ \epsilon = -1 }[/math], geometry) the relation gains a volume term, but only if the dimension n is even:[math]\displaystyle{ \sum_{F \subset P} (-1)^{\dim F} \angle(F) = \epsilon^{n/2}(1 + (-1)^n)\operatorname{Vol}(P) }[/math]Here, [math]\displaystyle{ \operatorname{Vol}(P) }[/math] is the normalized (hyper)volume of the polytope (i.e, the fraction of the n-dimensional spherical or hyperbolic space); the angles [math]\displaystyle{ \angle(F) }[/math] 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:[math]\displaystyle{ \sum_{v} \alpha_v - \sum_e \pi + 2\pi = 0 }[/math]where the first term [math]\displaystyle{ A=\textstyle\sum \alpha_v }[/math] is the sum of the internal vertex angles, the second sum is over the edges, each of which has internal angle [math]\displaystyle{ \pi }[/math], and the final term corresponds to the entire polygon, which has a full internal angle [math]\displaystyle{ 2\pi }[/math]. For a polygon with [math]\displaystyle{ n }[/math] faces, the theorem tells us that [math]\displaystyle{ A - \pi n + 2\pi = 0 }[/math], or equivalently, [math]\displaystyle{ A = \pi (n - 2) }[/math]. For a polygon on a sphere, the relation gives the spherical surface area or solid angle as the spherical excess: [math]\displaystyle{ \Omega = A - \pi (n - 2) }[/math].

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

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