Gram–Euler theorem
In geometry, the Gram–Euler theorem,[1] Gram-Sommerville, Brianchon-Gram or Gram relation[2] (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 be an -dimensional convex polytope. For each k-face , with its dimension (0 for vertices, 1 for edges, 2 for faces, etc., up to n for P itself), its interior (higher-dimensional) solid angle is defined by choosing a small enough -sphere centered at some point in the interior of and finding the surface area contained inside . Then the Gram–Euler theorem states:[3][1]
When the polytope is simplicial additional angle restrictions known as Perles relations hold, analogous to the Dehn-Sommerville equations for the number of faces.[2]
Examples[]
For a two-dimensional polygon, the statement expands into:
For a three-dimensional polyhedron the theorem reads:
History[]
The n-dimensional relation was first proven by Sommerville, Heckman and Grünbaum for the spherical, hyperbolic and Euclidean case, respectively.[2]
See also[]
- Euler characteristic
- Dehn-Sommerville equations
- Angular defect
- Gauss-Bonnet theorem
References[]
- ^ a b Perles, M. A.; Shepard, G. C. (1967). "Angle sums of convex polytopes". Mathematica Scandinavica. 21 (2): 199–218. doi:10.7146/math.scand.a-10860. ISSN 0025-5521. JSTOR 24489707.
- ^ a b c d Camenga, Kristin A. (2006). "Angle sums on polytopes and polytopal complexes". Cornell University.
- ^ Grünbaum, Branko (October 2003). Convex Polytopes. Springer. pp. 297–303. ISBN 978-0-387-40409-7.
- Polytopes
- Real algebraic geometry
- Geometry