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

For a three-dimensional polyhedron the theorem reads:$$\sum _{v}\Omega _{v}-2\sum _{e}\theta _{e}+\sum _{f}2\pi -4\pi =0$$where $\Omega _{v}$ is the solid angle at a vertex, $\theta _{e}$ 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 $2\pi$) and the last term is the interior solid angle (full sphere or $4\pi$).

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