A beautiful identity in Euclidean geometry is the Brianchon-Gram relation (also called the Gram-Sommerville formula, or Gram’s equation), which says the following: let be a convex polytope, and for each face of , let denote the solid angle along the face, as a fraction of the volume of a linking sphere. The relation then says:
Theorem (Brianchon-Gram relation): . In other words, the alternating sum of the (solid) angles of all dimensions of a convex polytope is zero.
Sketch of Proof: we prove the theorem in the case that is a simplex ; the more general case follows by generalizing to pyramids, and then decomposing any polytope into pyramids by coning to an interior point. This argument is due to Shephard.
Associated to each face is a spherical polyhedron in ; if the span of is the intersection of a family of half-spaces bounded by hyperplanes with inward normals , then is the set of unit vectors whose inner product with each is non-negative. Note further that for each there is some that pairs non-negatively with ; consequently to each one can assign a subset of indices, so that pairs non-negatively with if and only if . On the other hand, each subset determines a unique face of dimension . By the inclusion-exclusion formula, we conclude that “equals” zero, thought of as a signed union of spherical polyhedra. Since , the formula follows. qed.
Another well-known proof starts by approximating the polytope by a rational polytope (i.e. one with rational vertices). The proof then goes via Macdonald reciprocity, using generating functions.
Example: Let be a triangle, with angles . The solid angle at an interior point is , and the solid angle at each edge is . Hence we get and therefore in this case Brianchon-Gram is equivalent to the familiar angle sum identity for a triangle: .
Example: Next consider the example of a Euclidean simplex . The contribution from the interior is , and the contribution from the four facets is . There are six edges, with angles , that contribute . Each vertex contributes one spherical triangle, with (spherical) angles for certain , where each appears as a spherical angle in exactly two spherical triangles. The Gauss-Bonnet theorem implies that the area of a spherical triangle is equal to the angle sum defect: so the vertices contribute and the identity is seen to follow in this case too.
Note in fact that the usual proof of Gauss-Bonnet for a spherical triangle is done by an inclusion-exclusion argument involving overlapping lunes, that is very similar to the proof of Brianchon-Gram given above.
The sketch of proof above just as easily proves an identity in the spherical scissors congruence group. For equal to spherical, Euclidean or hyperbolic space of dimension , the scissors congruence group is the abelian group generated by formal symbols where and is a choice of orientation, modulo certain relations, namely:
- if the are contained in a hyperplane
- an odd permutation of the points induces multiplication by ; changing the orientation induces multiplication by
- if is an isometry of , then
- for any set of points, and any orientation
(Note that this definition of scissors congruence is consistent with that of Goncharov, and differs slightly from another definition consistent with Sah; this difference has to do with orientations, and has as a consequence the vanishing of spherical scissors congruence in even dimensions; whereas with Sah’s definition, for each )
The argument we gave above shows that for any Euclidean simplex , we have in .
Scissors congruence satisfies several fundamental properties:
- in . To see this, “triangulate” the sphere as a pair of degenerate simplices, whose vertices lie entirely on a hyperplane.
- There is a natural multiplication ; to define it on simplices, think of as the unit sphere in . A complementary pair of subspaces and intersect in a linked pair of spheres of dimensions ; if are spherical simplices in these subspaces, the image of is the join of these two simplices in .
It follows that the polyhedra in whenever is a face of dimension at least ; for in this case, is the join of a spherical simplex with a sphere of some dimension, and is therefore trivial in spherical scissors congruence. Hence the identity above simplifies to in .
One nice application is to extend the definition of Dehn invariants to ideal hyperbolic simplices. We recall the definition of the usual Dehn invariant. Given a simplex , for each face we let denote the spherical polyhedron equal to the intersection of with the link of . Then . Ideal scissors congruence makes sense for ideal hyperbolic simplices, except in dimension one (where it is degenerate). For ideal hyperbolic simplices (i.e. those with some vertices at infinity), the formula above for Dehn invariant is adequate, except for the -dimensional faces (i.e. the edges) . This problem is solved by the following “regularization” procedure due to Thurston: put a disjoint horoball at each ideal vertex of , and replace each infinite edge by the finite edge which is the intersection of with the complement of the union of horoballs; hence one obtains terms of the form in . This definition apparently depends on the choice of horoballs. However, if are two different horoballs, the difference is a sum of terms of the form where is constant, and ranges over the edges sharing the common ideal vertex. The intersection of with a horosphere is a Euclidean simplex , and the are exactly the spherical polyhedra as ranges over the vertices of . By what we have shown above, the sum is trivial in scissors congruence; it follows that is well-defined.
For more general ideal polyhedra (and finite volume complete hyperbolic manifolds) one first decomposes into ideal simplices, then computes the Dehn invariant on each piece and adds. A minor variation of the usual argument on closed manifolds shows that the Dehn invariant of any complete finite-volume hyperbolic manifold vanishes.
Update(7/29/2009): It is perhaps worth remarking that the Brianchon-Gram relation can be thought of, not merely as an identity in spherical scissors congruence, but in the “bigger” spherical polytope group, in which one does not identify simplices that differ by an isometry. Incidentally, there is an interesting paper on this subject by Peter McMullen, in which he proves generalizations of Brianchon-Gram(-Sommerville), working explicitly in the spherical polytope group. He introduces what amounts to a generalization of the Dehn invariant, with domain the Euclidean translational scissors congruence group, and range a sum of tensor products of Euclidean translational scissors congruence (in lower dimensions) with spherical polytope groups. It appears, from the paper, that McMullen was aware of the classical Dehn invariant (in any case, he was aware of Sah’s book) but he does not refer to it explicitly.