Topological Puzzles in the Marjan

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I encountered these interesting puzzles while hiking in the Marjan. They are made entirely of recycled materials and dead trees, and therefore cost nothing to build except a bit of labor. Essentially, they challenge the user to kinetically (i.e., by manipulating chains / one’s own body) calculate elements of the fundamental group of different shapes. It was amazing to see such a great example of public math education and engagement through puzzles in parks, especially in one as beautiful and historic as the Marjan! I have been in contact with the artist(who, sadly, is not a mathematician), and have received her permission to adapt the puzzles. I am hoping to find a broader community to engage in this sort of work and perhaps construct similar puzzles in the Southern New England area. If you are interested in this sort of thing, please get in touch with me!

The New Fountain Manifesto

I’d love to get paid to design fountains. It’s because I find most of the ones I see very boring. The vast majority of fountains in Western culture are based on pressure-driven flows, which basically gives rise to a one-parameter family of shapes (jets of a certain radius) in the actual fluid on display. Sometimes they’re made to look fancy by arranging the jets themselves into particular shapes or emerging from attractive fixtures, but the flow itself — and the information conveyed by the flow — is very boring. Pressure gradients inherently lack the natural aesthetic of other types of fluid mechanics. In the absence of turbulent features (which change the opacity of the fluid, leading to whitewater and other minor visual effects), the human eye can only detect flow properties in a thin film near the surface of the fluid bulk, and is not very adept at gauging relative flow speeds. When we look into a river, the visual cues are often in fluid-structure interaction: fish, kelp, sediment. Furthermore, the human eye does not have much of a mechanism for recognizing continuum stress.

I have been working with designer and fellow fluid mechanician Joseph Burg from Berkeley on several fountains which bypass this rather unfortunate monopoly of laminar pressure-driven flows. We try to put non-Newtonian rheology, optics of liquid crystals and Marangoni flows, and fluid-elastic interactions on display. You can find some of our work here. If you are interested in discussing these kinds of ideas, or suggesting/commissioning some work of this time, please don’t hesitate to drop Joe or me a line!

Pattern Formation and Catastrophe Optics in Lighting Design

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!

Compendium of links for topological puzzles

I have been entertaining the notion of designing short educational modules based on topological puzzles. I have roughly the 4th-8th grade range in mind. Here I collect various links and PDF’s which could be used as material for such an endeavor:

A nice history of some easily-constructed puzzles

Large collection of topological puzzles

The Wu riddles – a collection of mathematical riddles, including some of a topological and algebraic character