I learned a cool Euclidean geometry fact this week in our colloquium, in a talk by Benoît Bertrand of the University of Toulouse on joint work of his with my UNAM colleague Lucía López de Medrano. They don't think this geometry fact can be new, but also haven't been able to find a reference. The theorem is a generalization of the fact that the sum of the angles of a triangle is 180° to higher dimensions.

This generalization is not that the sum of the solid angles of a tetrahedron is constant: that's not true! If you think of tetrahedra that are almost flat, that is, whose vertices are nearly coplanar, there are two types: (1) the vertices are nearly the vertices of a convex planar quadrilateral, or (2) the vertices are nearly a triangle with a point inside it. In case (1) the solid angles are close to 0; in case (2) the one at the central vertex is close to half a sphere (2π steradians), and the others are close to 0. It turns out that those are the extremes and the sum of the solid angles of a tetrahedron is always between 0 and 2π steradians.

So how does their generalization go? Consider an n-dimensional simplex and select one of its vertices, say P. Now consider all ways coloring the vertices of the simplex with either red or blue in such a way that P is blue (this is half of all colorings). For each coloring look at the vectors of the form R-B where R is some red vertex and B is a blue one, and take the cone they generate (that is, the linear combinations of those vectors with non-negative coefficients). Their theorem says that those cones tile a half-space! (Here "tile" means the interiors are disjoint and the union of the cones is a half-space.)

Next, a proof. 1/2