Note on an n-dimensional Pythagorean theorem

A famous theorem in Euclidean geometry often attributed to the Greek thinker Pythagoras of Samos (6th century, B.C.) states that if one of the angles of a planar triangle is a right angle, then the square of the length of the side opposite the right angle equals the sum of the squares of the lengths of the sides which form the right angle. There are less commonly known higherdimensional versions of this theorem which relate the areas of the faces of a simplex having one “orthogonal vertex” by analogous sums-of-squares identities. In this note I state and prove one such result, hoping that students of mathematics will become better acquainted with it. Introduction “Geometry has two great treasures. One is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.” J. Kepler ([2], p. 58.) In high school geometry one learns the Pythagorean theorem, stating that “the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides” (see [6], Propositions 47 and 48). This fact was known to the Babylonians over one thousand years prior to the time of Pythagoras, as numerous clay tablets such as the famous Plimpton 322 tablet show [2], [8], [13], [16]. In fact, [4] contains a reference to a tablet dating back to 2600 B.C. on which an illustrated example of the Pythagorean theorem is given (curiously, there is no evidence suggesting that the ancient Egyptians were aware of the theorem, [7]). Hundreds of proofs of this beautiful fact are known; see for example the reference [12]. There is little doubt that the theorem is one of the most fundamental of mathematical facts. Recently, I was beholding the wonder of this result, the fact that the mere requirement of orthogonality of one of the angles implies that such a simple quadratic relation must be satisfied by the lengths of the sides, and it occurred to me that it would be no more surprising if an analogous result held in higher dimensions. This turned out to be true, and is the subject of the present note. After showing the generalized result to several colleagues and asking them if they could provide a reference for a fact which should obviously be widely known, it became clear to me that, incredibly, comparatively few mathematicians are aware of it. Boyer ([2], 14.10, p. 288) attributes a three-dimensional generalization of the classical Pythagorean theorem to Fibonacci in his early 13th century work Practica Geometriae. Guggenheimer [9] suggests that the n-dimensional result is due to J. P. de Gua de Malves, in Histoire de l’Académie des Sciences pour l’année 1783, p. 375, Paris, 1786. The n-dimensional statement that I will present below is in fact a special case of a result described by Monge and Hachette in an early 19th century treatise on analytic geometry (see [2], 22.8, p. 533). Various versions of the result have been found anew numerous times; a Web search returned, among others, the papers [1], [3], [5], [10], [11], [14], [15]. I hope that the present note will further promote increased recognition of this beautiful and simple result.