Sums of Two Squares Visualized

In 1972, John Brillhart described an algorithm for expressing a prime p ≡ 1 (mod 4) as the sum of two squares. Brillhart's algorithm, which is based on the Euclidean algorithm, is simplicity itself. However, Brillhart's proof of his algorithm's correctness uses several previous results, and subsequent simplifications of his argument still retain something of an air of mystery. We provide a geometric interpretation of Brillhart's algorithm, which not only proves that it works, but also sheds light on a surprising palindromic property proved by Perron, and used by Brillhart in his proof.