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Shapes and chance

Light and Rainbows

Rays of light followed exactly through a drop, a bead, a pebble or a prism, drawn in 3D, with every number held by tests. Each ray bends by Snell's law at every face and splits between reflecting and passing as Fresnel's equations say, with the index each colour meets: crown glass from SCHOTT's N-BK7 datasheet, whose nd of 1.51680 and Abbe number of 64.17 the tests get back, fused silica from Malitson's 1965 fit, and water from a five-line table at 20 C fitted with Cauchy's formula. The tests read Snell's law back from every refracted ray, check that no light is lost, that a sphere never traps a ray, that the traced bow lands on Descartes' formula for every colour, red 42.3 and violet 40.6 degrees in the primary and 50.4 to 53.5 in the secondary with Alexander's dark band between, that a glass bead's bow sits near 21 degrees, and that a prism turns light least on the symmetric path, by the closed-form angle. The sky panel adds up hundreds of thousands of traced rays and colours them as the eye sees, by a fit to the CIE 1931 colour matching functions. Ours: every shape and size, white light with the same power at every colour, and the beam's width on the prism's screen. Rays cannot show the wave effects, the faint bands just inside a real bow.