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 $\mathrm{\angle }(F)$ 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: $$\sum _{F\subset P}(-1{)}^{\dim F}\mathrm{\angle }(F)=0$$In non-Euclidean geometry of constant curvature (i.e. spherical, $ϵ=1$, and hyperbolic, $ϵ=-1$, geometry) the relation gains a volume term, but only if the dimension n is even:$$\sum _{F\subset P}(-1{)}^{\dim F}\mathrm{\angle }(F)={ϵ}^{n/2}(1+(-1{)}^n)\mathrm{Vol}(P)$$Here, $\mathrm{Vol}(P)$ is the normalized (hyper)volume of the polytope (i.e, the fraction of the n-dimensional spherical or hyperbolic space); the angles $\mathrm{\angle }(F)$ 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 $A=\sum {\alpha }_v$ 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: $\mathrm{\Omega }=A-\pi (n-2)$.

For a three-dimensional polyhedron the theorem reads:$$\sum _v{\mathrm{\Omega }}_v-2\sum _e{\theta }_e+\sum _{f}2\pi -4\pi =0$$where ${\mathrm{\Omega }}_v$ 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