I have long been interested in the application of singularity/”catastrophe” theory (I use quotation marks because, like V.I. Arnold, I do not like the sensationalist overtones of “catastrophe theory”) to the design of lighting applications using natural lenses and natural light. The normal forms of caustics in geometric optics are well-known and you can find pictures of them in various places. The most useful relationship in my opinion is that given e.g. in Berry for the distance between a fluid lens in the shape of a spherical cap and the focal set of the resulting elliptic umbilic caustic when light penetrates normally to the north pole of the cap. Because of the symmetry of the liquid lens, there is a one-one relationship between the radius of the cap and the distance of the nearest bright spot. It is a tricky problem to protect a liquid lens from perturbations to its curvature, but this is done all the time in microfluidic systems — the only design problem is one of scale. If it is feasible it is also a great way to incorporate natural light in difficult situations, such as in the subway. Fluids and lighting also have a natural aesthetic.

If you have either money or design skills and are interested in this sort of thing, please let me know!

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