MATRICES THAT DEFINE SERIES OF PYTHAGOREAN TRIPLES THAT HAVE A TRIANGLE WITH ONE IRRATIONAL SIDE AS LIMIT

Abstract

Making use of the universal set of Pythagorean triples, series of triples are defined where triple n + 1 is obtained by multiplying triple n with a specific 3 × 3 matrix. In terms of Pythagorean triangles, the shape of the limiting triangle in these series is a triangle with one of its sides having an irrational ratio with respect to the other sides. These specific matrices may be directly associated with the square roots of uneven positive integers (that are not perfect squares), and also some of the even positive integers since the limit of the powers of these matrices, applied to any 3 × 1 matrix of real numbers leads to a specific right-angled triangle that contains that square root as one of its sides.