2008/09/19 by David Feldman, David V. Feldman, Feldman, David V. +2
Mathematics · #52B11 #FOS: Mathematics #Metric Geometry (math.MG) #Statistics Education and Methodologies #math.MG #msc:52B11
paper · pdf · doi:10.48550/arxiv.0809.3459
4 pages
arxiv created 2008/09/19 · openalex publication_date 2008/09/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.