Visualizing Integer Length Integer Vectors with J and POV-Ray

摘要

Integer length integer vectors correspond to solutions of Diophantine equations where a sum of squares is a square. In two dimensions, this equation is a~2 + b~2 = c~2, and solutions in positive integers correspond to Pythagorean Triples (or integer length integer vectors < a, b > in Z~2). In three dimensions, this equation is x~2 + y~2 + z~2 = r~2, and integer solutions correspond to integer length integer vectors < x, y, z > in Z~3.