With the high school mathematics curriculum as full as it is, why should we busy teachers take class time away from the syllabus to investigate patterns of no discernible practical value? My answer is that students should be made aware that there is more to mathematics than learning how to solve an endless variety of problems. There is a beauty in its patterns, and especially for those who may not pursue mathematics beyond high school, these patterns may be the only mathematics they will remember. Recently in my geometry class we discovered a pattern in Pythagorean triples that, in addition to being interesting in itself, can be used at various levels to illustrate inductive reasoning, proof by mathematical induction, and other algebraic proofs. It can also generate several different computer programs.