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[edit]
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]
![{\displaystyle \sum _{F\subset P}(-1)^{\dim F}\angle (F)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54548e16a5c4278cf3fe4321cd1524b3b6e3da4f)
In
non-Euclidean geometry of constant curvature (i.e.
spherical,
![{\displaystyle \epsilon =1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6cea33fb1374a6510eb3a7062b52245de1b33ee)
, and
hyperbolic,
![{\displaystyle \epsilon =-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b62b368e6c6328360a26ea2c34d5d94e917ce7be)
, geometry) the relation gains a volume term, but only if the dimension
n is even:
![{\displaystyle \sum _{F\subset P}(-1)^{\dim F}\angle (F)=\epsilon ^{n/2}(1+(-1)^{n})\operatorname {Vol} (P)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e61eac9efafe2ef680ca1c042e5ea3a0170a57)
Here,
![{\displaystyle \operatorname {Vol} (P)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/249e8a411133ab1bab16d944a1fed6258f73252e)
is the normalized (hyper)volume of the polytope (i.e, the fraction of the
n-dimensional spherical or hyperbolic space); the angles
![{\displaystyle \angle (F)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0e1dd32658594c788b20dca95e154df9d25bc5)
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[edit]
For a two-dimensional polygon, the statement expands into:
![{\displaystyle \sum _{v}\alpha _{v}-\sum _{e}\pi +2\pi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74fcf1589a78946d7bc215526473a3ad3a48dc90)
where the first term
![{\displaystyle A=\textstyle \sum \alpha _{v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cee0fcb39151e0a11155803963d1ea776f7321)
is the sum of the internal vertex angles, the second sum is over the edges, each of which has internal angle
![{\displaystyle \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
, and the final term corresponds to the entire polygon, which has a full internal angle
![{\displaystyle 2\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06)
. For a polygon with
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
faces, the theorem tells us that
![{\displaystyle A-\pi n+2\pi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d229a935b5def3bc5aa5f689c293ad61be5f345)
, or equivalently,
![{\displaystyle A=\pi (n-2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e118ec3e19e764d9cbfd698709dcb7876a111a92)
. For a polygon on a sphere, the relation gives the spherical surface area or
solid angle as the
spherical excess:
![{\displaystyle \Omega =A-\pi (n-2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084cb946d7ac2987c508ed8b1415816104782882)
.
For a three-dimensional polyhedron the theorem reads:
![{\displaystyle \sum _{v}\Omega _{v}-2\sum _{e}\theta _{e}+\sum _{f}2\pi -4\pi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02d0700f12b5411c94411d3dd4efd8a43bc66dc9)
where
![{\displaystyle \Omega _{v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d6f1be675b12ad16e7d9ac143acbe88af2b5d8a)
is the solid angle at a vertex,
![{\displaystyle \theta _{e}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/930299aaf5570dad3cfeb4dbca8d9623e9d739cd)
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
![{\displaystyle 2\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06)
) and the last term is the interior solid angle (full sphere or
![{\displaystyle 4\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/057444bf35a0c22b19bcae1ef06e06ecdf8abe56)
).
History[edit]
The n-dimensional relation was first proven by Sommerville, Heckman and Grünbaum for the spherical, hyperbolic and Euclidean case, respectively.[2]
See also[edit]
References[edit]