This Demonstration plots primitive Pythagorean triples (PPTs) to make lovely, interesting patterns in the - plane. The details provide definitions to classify all triples that are solutions of the Pythagorean formula in integers. The 2D plot shows integer points that satisfy this equation for some integer .

Let be a Pythagorean triple (PT), that is, , , and are positive integers such that . A primitive Pythagorean triple (PPT) is a PT with . Redundants are solutions that allow , , and to be negative. This Demonstration illustrates all 16 solutions, and .

Click "PPT 1" or "PPT 2" to see the subset of PPTs with base-3 ID numbers beginning with 1 or 2.

If we interpret the plotted locations as Gaussian integers, then selecting the checkbox plots the square root of those locations. This square root mode shows why Euclid's formula only appies to "PPT 1" primitive Pythagorean triples.