Let be an -dimensional convexpolytope. 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]
Examplesedit
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 ).
Historyedit
The n-dimensional relation was first proven by Sommerville, Heckman and Grünbaum for the spherical, hyperbolic and Euclidean case, respectively.[2]
gram, euler, theorem, geometry, gram, sommerville, brianchon, gram, gram, relation, named, after, jørgen, pedersen, gram, leonhard, euler, duncan, sommerville, charles, julien, brianchon, generalization, internal, angle, formula, polygons, higher, dimensional,. In geometry the Gram Euler theorem 1 Gram Sommerville Brianchon Gram or Gram relation 2 named after Jorgen 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 Contents 1 Statement 2 Examples 3 History 4 See also 5 ReferencesStatement editLet P displaystyle P nbsp be an n displaystyle n nbsp dimensional convex polytope For each k face F displaystyle F nbsp with k dim F displaystyle k dim F nbsp its dimension 0 for vertices 1 for edges 2 for faces etc up to n for P itself its interior higher dimensional solid angle F displaystyle angle F nbsp is defined by choosing a small enough n 1 displaystyle n 1 nbsp sphere centered at some point in the interior of F displaystyle F nbsp and finding the surface area contained inside P displaystyle P nbsp Then the Gram Euler theorem states 3 1 F P 1 dim F F 0 displaystyle sum F subset P 1 dim F angle F 0 nbsp In non Euclidean geometry of constant curvature i e spherical ϵ 1 displaystyle epsilon 1 nbsp and hyperbolic ϵ 1 displaystyle epsilon 1 nbsp geometry the relation gains a volume term but only if the dimension n is even F P 1 dim F F ϵ n 2 1 1 n Vol P displaystyle sum F subset P 1 dim F angle F epsilon n 2 1 1 n operatorname Vol P nbsp Here Vol P displaystyle operatorname Vol P nbsp is the normalized hyper volume of the polytope i e the fraction of the n dimensional spherical or hyperbolic space the angles F displaystyle angle F nbsp 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 editFor a two dimensional polygon the statement expands into v a v e p 2 p 0 displaystyle sum v alpha v sum e pi 2 pi 0 nbsp where the first term A a v displaystyle A textstyle sum alpha v nbsp is the sum of the internal vertex angles the second sum is over the edges each of which has internal angle p displaystyle pi nbsp and the final term corresponds to the entire polygon which has a full internal angle 2 p displaystyle 2 pi nbsp For a polygon with n displaystyle n nbsp faces the theorem tells us that A p n 2 p 0 displaystyle A pi n 2 pi 0 nbsp or equivalently A p n 2 displaystyle A pi n 2 nbsp For a polygon on a sphere the relation gives the spherical surface area or solid angle as the spherical excess W A p n 2 displaystyle Omega A pi n 2 nbsp For a three dimensional polyhedron the theorem reads v W v 2 e 8 e f 2 p 4 p 0 displaystyle sum v Omega v 2 sum e theta e sum f 2 pi 4 pi 0 nbsp where W v displaystyle Omega v nbsp is the solid angle at a vertex 8 e displaystyle theta e nbsp 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 p displaystyle 2 pi nbsp and the last term is the interior solid angle full sphere or 4 p displaystyle 4 pi nbsp History editThe n dimensional relation was first proven by Sommerville Heckman and Grunbaum for the spherical hyperbolic and Euclidean case respectively 2 See also editEuler characteristic Dehn Sommerville equations Angular defect Gauss Bonnet theoremReferences edit 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 arXiv math 0607469 Grunbaum Branko October 2003 Convex Polytopes Springer pp 297 303 ISBN 978 0 387 40409 7 Retrieved from https en wikipedia org w index php title Gram Euler theorem amp oldid 1140531209, wikipedia, wiki, book, books, library,