Tan Halley Scallop

Halley's method applied to f(z) = tan z - z, with the step collapsed to T(T - z) / (z(1 + T^2) - T) where T = tan z, so the poles at (k + 1/2) pi stay finite. Two root families split the palette. The triple root at the origin converges linearly and takes the sapphire ground, and every root of tan r = r converges cubically and takes an ice lens. The window sits on the lens around the root and pole near 5 pi / 2, and the faint rings are one per Halley step.