Английская Википедия: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 <math>P</math> be an <math>n</math>-dimensional convex polytope. For each k-face <math>F</math>, with <math>k = \dim(F)</math> its dimension (0 for vertices, 1 for edges, 2 for faces, etc., up to n for P itself), its interior (higher-dimensional) solid angle <math>\angle(F)</math> is defined by choosing a small enough <math>(n - 1)</math>-sphere centered at some point in the interior of <math>F</math> and finding the surface area contained inside <math>P</math>. Then the Gram–Euler theorem states:[3][1] <math display="block">\sum_{F \subset P} (-1)^{\dim F} \angle(F) = 0</math>In non-Euclidean geometry of constant curvature (i.e. spherical, <math>\epsilon = 1</math>, and hyperbolic, <math>\epsilon = -1</math>, geometry) the relation gains a volume term, but only if the dimension n is even:<math display="block">\sum_{F \subset P} (-1)^{\dim F} \angle(F) = \epsilon^{n/2}(1 + (-1)^n)\operatorname{Vol}(P)</math>Here, <math>\operatorname{Vol}(P)</math> is the normalized (hyper)volume of the polytope (i.e, the fraction of the n-dimensional spherical or hyperbolic space); the angles <math>\angle(F)</math> 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:<math display="block">\sum_{v} \alpha_v - \sum_e \pi + 2\pi = 0</math>where the first term <math>A=\textstyle\sum \alpha_v</math> is the sum of the internal vertex angles, the second sum is over the edges, each of which has internal angle <math>\pi</math>, and the final term corresponds to the entire polygon, which has a full internal angle <math>2\pi</math>. For a polygon with <math>n</math> faces, the theorem tells us that <math>A - \pi n + 2\pi = 0</math>, or equivalently, <math>A = \pi (n - 2)</math>. For a polygon on a sphere, the relation gives the spherical surface area or solid angle as the spherical excess: <math>\Omega = A - \pi (n - 2)</math>.
For a three-dimensional polyhedron the theorem reads:<math display="block">\sum_{v} \Omega_v - 2\sum_e \theta_e + \sum_f 2\pi - 4\pi = 0</math>where <math>\Omega_v</math> is the solid angle at a vertex, <math>\theta_e</math> 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 <math>2\pi</math>) and the last term is the interior solid angle (full sphere or <math>4\pi</math>).
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
