A New Approach with Software Implementation to Extend the Pythagorean Theorem in Multi-Dimensions

Abstract

The Pythagorean theorem, which states that "for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the two legs," applies only within a single plane (i.e., in two dimensions). This paper proposes an extension of the Pythagorean theorem to multiple planes and dimensions by introducing a novel concept called Abd's 1st Intuition or Abd's First Theory (AFT). In this extended version, a finite sequence of right triangles is constructed, starting from a specific initial right triangle with two legs and a hypotenuse. Each subsequent right triangle is formed with one leg as the hypotenuse of the previous triangle and the other leg as a new value. This process continues such that the n-th right triangle has legs corresponding to the previous hypotenuse and a new dimension's value, resulting in a new hypotenuse. Each transition to a new plane introduces a new dimension, culminating in an extended form of the theorem: the square of the hypotenuse of this final right triangle is equal to the sum of the squares of the legs from all the constructed triangles. Additionally, a sub-exponential growth pattern for the sequence of hypotenuses is derived, demonstrating that their growth rate depends on the acute angle adjacent to the legs. For the special case of isosceles right triangles, the hypotenuse of the n-th triangle can be explicitly calculated from the first leg of the base triangle. This generalized theorem is intended for applications in wireless and wired communication networks, cryptography, communications security, and other areas within applied programming and computer science.