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This is points P_k at polar coords (radius = k·r, angle = k·θ) as θ spins. Connect sequentially: • small θ → smooth Archimedean spiral • θ = 2π·p/q → perfect n-gons & stars emerge!

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Mathematica

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n₁ sin(θ₁) = n₂ sin(θ₂) This elegant equation describes how light bends at the boundary between two media. Here, n₁ and n₂ are the refractive indices, while θ₁ and θ₂ are the angles of incidence and refraction measured from the surface normal. As light rays obey this simple trigonometric relationship, beautiful caustic curves emerge and focal points shift with every change in geometry or material. It is pure geometric optics at its finest: the raw interaction of light and matter, reminding us that the universe is governed by beautiful, predictable symmetries. This is used while designing high-resolution camera lenses (correcting aberrations for sharp focus) and enabling fiber-optic communication through total internal reflection.
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