Greg Egan on Nostr: How can we generalise the fact that the angles at the vertices of a triangle add up ...

How can we generalise the fact that the angles at the vertices of a triangle add up to 180°, to apply to tetrahedra and higher-dimensional n-simplexes?

Here’s a result discovered by Benoît Bertrand and Lucía López de Medrano that I learned from Omar Antolín.

Pick one vertex of the n-simplex, colour it blue, then consider all 2^n-1 ways to colour the other vertices red or blue, omitting the all-blue colouring.

For each such colouring, collect all the vectors that point from a blue vertex to a red vertex, and form their cone: the set of linear combinations of these vectors with positive coefficients.

Then these 2^n-1 cones will all fit together to form a half-space!

This is what we get for triangles.