Gram–Euler theorem explained

In geometry, the Gram–Euler theorem, Gram-Sommerville, Brianchon-Gram or Gram relation^[1] (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:^{[2] [3]} $$\backslash sum\_\; (-1)^\; \backslash angle(F)\; =\; 0$$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:$$\backslash sum\_\; (-1)^\; \backslash angle(F)\; =\; \backslash epsilon^(1\; +\; (-1)^n)\backslash operatorname(P)$$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:$$\backslash sum\_\; \backslash alpha\_v\; -\; \backslash sum\_e\; \backslash pi\; +\; 2\backslash pi\; =\; 0$$where the first term

A = style\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:$$\backslash sum\_\; \backslash Omega\_v\; -\; 2\backslash sum\_e\; \backslash theta\_e\; +\; \backslash sum\_f\; 2\backslash pi\; -\; 4\backslash pi\; =\; 0$$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

References

1. Camenga. Kristin A.. 2006. Angle sums on polytopes and polytopal complexes. Cornell University. math/0607469 .
2. Book: Grünbaum, Branko. Convex Polytopes. Convex Polytopes. October 2003. Springer. 978-0-387-40409-7. 297–303.
3. Perles. M. A.. Shepard. G. C.. 1967. Angle sums of convex polytopes. Mathematica Scandinavica. 21. 2. 199–218. 10.7146/math.scand.a-10860. 24489707. 0025-5521. free.