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]

In non-Euclidean geometry of constant curvature (i.e. spherical, , and hyperbolic, , geometry) the relation gains a volume term, but only if the dimension n is even:

Here, is the normalized (hyper)volume of the polytope (i.e, the fraction of the n-dimensional spherical or hyperbolic space); the angles also have to be expressed as fractions (of the (n-1)-sphere).[2]

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:

where the first term is the sum of the internal vertex angles, the second sum is over the edges, each of which has internal angle , and the final term corresponds to the entire polygon, which has a full internal angle . For a polygon with faces, the theorem tells us that , or equivalently, . For a polygon on a sphere, the relation gives the spherical surface area or solid angle as the spherical excess: . For a three-dimensional polyhedron the theorem reads:

where is the solid angle at a vertex, 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 ) and the last term is the interior solid angle (full sphere or ).

History

The n-dimensional relation was first proven by Sommerville, Heckman and Grünbaum for the spherical, hyperbolic and Euclidean case, respectively.[2]

See also

References

  1. 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.
  2. Camenga, Kristin A. (2006). "Angle sums on polytopes and polytopal complexes". Cornell University. arXiv:math/0607469.
  3. Grünbaum, Branko (October 2003). Convex Polytopes. Springer. pp. 297–303. ISBN 978-0-387-40409-7.
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