Pythagoras Reconceived: A Perspective With Regard To the Relationship between Discrete Points in Space
Pythagoras' Theorem states that x2 + y2 = z2 where x, y, and z are the
distance between three points (e.g. A, B, & C) on a plane and where ?ABC is a right angle. The Theorem is limited to two dimensions as shown by Wiles (Annals Math. (1995) 142: 443-551). The problem as to "why" the equation is so has not been asked nor answered until recently. Here it is suggested that the square of the distance between two points on a plane represents the relationship between those two points (for non-right triangles the equation x2 + y2 - 2xy.cos? = z2 is used). This is exemplified by the equations relating gravity (Fgrav = (G.m1.m2)/d2), energy and mass (Ek = ½ mv2; E = mc2), electrical power, resistance, current, and voltage (P = I2 R; P = V2/R); and algorithms used in statistical analyses (Alon et al. (1999) Proc. Natl. Acad. Sci. 96: 6745-6750). For dimensions >2, distances can be constructed from two-dimensions in vector format. Many other relationships in nature are now open for interdisciplinary study, including the chemistry of biology at all levels from atomic interactions within molecules to the inter-relationships between organisms, psychology, economics, sociology, and philosophy.