Why does a bright ring of radius ≈22° so often circle the Sun or Moon through thin cloud — the same size every time, whatever the ice? And why is it red on the inside, the reverse of a rainbow?
▶ Launch the interactive simulationTrace a ray through a hexagonal ice crystal as a 60° prism: refract in (Snell r1 = asin(sin i / n)), cross to the far face at r2 = A − r1, refract out (e = asin(n sin r2)), and record the deviation D(i) = i + e − A. Sweep the incidence i over every ray the tumbling crystals present and find the minimum of D(i) by a raw numeric scan + parabolic refine — never the closed form 2·asin(n sin(A/2)) − A, never assuming symmetric passage. Repeat for a 90° prism (the 46° halo), for red and blue ice indices (the colour split), and for the reflection rival D = 180° − 2θ.
BRAVAIS' HALO (1847) — D(i) has a MINIMUM, so tumbling ice crystals pile light up at the smallest deviation: a caustic. That bright ring sits at D_min = 21.84° for ice (n = 1.31), dark inside, INDEPENDENT of crystal size — recovered here to rel 3e-5 from a raw Snell sweep with no formula coded, the symmetric passage i = e emerging on its own. The SAME machinery on a 90° prism gives the 46° halo (45.7°), and a prism steeper than 2·asin(1/n) = 99.5° is fully total-internal-reflecting — why 46° is the largest. Dispersion puts blue at larger deviation, so the ring is RED on its INNER edge, the reverse of the rainbow's red-outer order. The naive 'halo = reflected sunlight' is falsified structurally: reflection D = 180° − 2θ has |dD/dθ| = 2 everywhere — no stationary point, no ring. The refraction sibling of ?world=rainbow's reflection caustic.