Pythagorean Descent

where B(v, w) = 12(Q(v + w) − Q(v) − Q(w)) is the bilinear form associated to Q. The transformation sw is linear, fixes the plane w⊥ = {v : v ⊥ w}, and acts by negation on the line through w. These properties characterize sw. We will use reflections associated to the four vectors e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1), and e1 + e2 + e3 = (1, 1, 1). The vectors e1, e2, and e3 form an orthogonal basis for Q and the three reflections se1 , se2 , and se3 are all in OQ(Z). Starting with an integral vector (a, b, c), the vectors (±a,±b,±c) are obtained from it by applying these three reflections. After making suitable sign changes, any integral null vector (a, b, c) can be assumed to have nonnegative coefficients, so 0 ≤ a, b ≤ c because c2 = a2 + b2. Assume a and b are both positive. Then