Angles as probabilities

TL;DR

This paper introduces a probabilistic interpretation of solid angles to relate the sum of inner angles of simplices to the probability that their orthogonal projections are lower-dimensional simplices, extending classical geometric facts.

Contribution

It generalizes the angle sum property of triangles to higher-dimensional simplices using a probabilistic framework based on solid angles.

Findings

Sum of solid inner angles divided by 2*pi equals projection probability for tetrahedra.

Generalizes to n-simplices with probability related to projections onto hyperplanes.

Briefly discusses applications to polytopes and classical Gram-Euler relations.

Abstract

We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a tetrahedron T, divided by 2*pi, gives the probability that an orthogonal projection of T onto a random 2-plane is a triangle. More generally, it is shown that the sum of the (solid) inner vertex angles of an n-simplex S, normalized by the area of the unit (n-1)-hemisphere, gives the probability that an orthogonal projection of S onto a random hyperplane is an (n-1)-simplex. Applications to more general polytopes are treated briefly, as is the related Perles-Shephard proof of the classical Gram-Euler relations.