A Comprehensive Pythagorean Theorem for All Dimensions

Abstract

We use a 200-year-old theorem on determinants to prove a very general version of the Pythagorean theorem. It relates the square of the n-dimensional volume of an n-parallelotope in ℝm to the sum of the squares of the n-dimensional volumes of the orthogonal projections of the parallelotope onto the k-dimensional coordinate subspaces of ℝm.