Dan’s Geometrical Curiosities

My friend Britta showed me this internet gem: a musical composition game where you make sound via the neighborhood. Isle of Tune allows you to build roadways decorated with trees, flowers, streetlamps and houses, each with its own characteristic sound that plays when one of up to three cars pass it. I sat down for a few hours and created the following tune. After showing it to some friends and getting positive feedback, I decided to notate a more popular song.

After attempting to fill every square with an object, I also managed to break Isle of Tune’s sidewalk building algorithm in this island. The gray sidewalks on the side of the roads appear bizarre and the cars also don’t go in the intended direction. I’m not sure whether I’m pleased with this, because the original island I had was pretty cool: it was a standard neighborhood layout with grid-like streets, and I even set up the arrows so that the cars would navigate back and forth across the whole island. On the other hand, it’s kind of interesting to see the result in this peculiar form, too.

All fun aside, I also wanted to make an Island that showed how the game could also be used to make more mathematically viable creations. A road is a progression in time (a stave, if you will), and items on the road are notes, representing different rhythms, pitches and timbres. A “city block” in this game can represent a musical loop, provided that the arrows on the corners are facing the right way to carry the car around the block. Square loops of different sizes will create different loop lengths. This immediately made me think of Steve Reich’s Clapping Music. Naturally, I had to implement that, too, in Isle of Tune.

Boxes are usually shaped like cubes, or rectangular prisms, at least. The challenge I set for myself was to build a box that not only reflected some more interesting mathematics and symmetries, but also presented itself in a way that an artistic-minded individual would be able to appreciate.

In the Winter of my Sophomore year at UCSB, I gathered information to find where the woodshop was on Campus. I did some mathematical calculations, and I determined the size and all the angles I would need to build the icosahedron box from wood.

Here is a photograph of my process of gluing the triangular pieces of the box together with the help of clamps and foam. It was very difficult to make sure that the joints were under enough compression that they would adhere, but not so much that it would warp the wood. I always had to keep in mind the global geometry of the project as opposed to looking edge by edge. If I had put too much of the box together, I may have realized that the last few pieces were unable to fit. It was a constant struggle to check dimensions and make sure the clamps had even pressure and weren’t distorting the box. With care, I continued gluing pieces until I had only one more of the twenty equilateral triangles unglued. This would form the opening of the box. Remarkably, this triangle was able to fit, so I drilled a hole in the centroid of it so that I could fasten a handle so the box could be opened easily.

I had to decide how I wanted to paint the box, and this brings me to my second mathematical aspect of this project: the reaction-diffusion system. I had recently been enamored by a fascinating partial differential equation— two spatial dimensions and one temporal dimension, and the equations look something like in the figure.

The Reaction-Diffusion system was designed to model the behavior of certain chemicals in interaction with each other, where U and V represent the amounts of chemicals A and B respectively at a certain point. You can learn more about this kind of Reaction-Diffusion system here or by reading about it on Wikipedia. They are a classic example of how complicated structures can emerge from very simple rules. I wanted to apply this pattern to more exotic geometry like my icosahedron. In the link I provided, the equation is solved on a surface of toroidal topology, that is, the system is solved on a square with opposite sides pairwise identified. However, the geometry of this torus is trivial because it is equivalent to solving the system on a square with periodic boundary conditions. I wanted to examine evolution of the system on a more complicated manifold, and the icosahedron was a perfect opportunity to experiment with this.

Since I already had implemented the Reaction-Diffusion solver into Mathematica, I only had to program the icosahedron geometry into Mathematica properly. I paired up the faces of the icosahedron into ten imaginary parallelogram-shaped regions, each of which can be converted into a square by a simple linear transformation. By computing this transformation, I was able to rewrite the Reaction-Diffusion equations using my new coordinates. With this technique, the program would run with ten square regions reacting differently, but simultaneously. When they were all ready, the pattern on each square would appear warped because I had put in a different equation via the new coordinates. Because I had done the math carefully, when I skewed the squares to parallelograms with the corner angle as 60 degrees in photoshop, the skewness would disappear, leaving exactly the desired pattern, but with the right diamond shape. You can see the skewing procedure in the figure showing the before and after pictures, and the unsightly skew effect that the square version has before it is transformed. The final icosahedron would have this effect if I did not change coordinates first.

The biggest challenge was to get the matching to work out between neighboring diamond panels; I manually typed into Mathematica which edges were adjacent to which other edges so that the boundary conditions were well defined. The icosahedron has 30 edges, but only 20 identifications of pairs of edges on the 10 diamond shaped panels were necessary, since 10 identifications were already performed in using diamonds instead of individual triangles.

Once all of the math was done, and my laptop had finished cranking numbers, I used Photoshop’s “Threshold” adjustment option, along with some effects like “Find Edges”. This gave me a traceable outline pattern that would help me paint the box later. After printing out all ten of these patterns, I painted my box yellow, and bought some carbon copy paper. This allowed me to trace the pattern right onto my box as a guide for painting the next layer. I painted with green inside the boundary of the squiggles, and finally screwed on the metal knob.

You may have seen it before: a magician spreads out a deck of cards on a table into a long row. He tilts the endmost card upward, and the neighboring cards rise accordingly, resting on the adjacent card. The next thing you know there is a beautiful shape, an array of cards coming to a sharp point which can be moved back and forth with the magician’s finger.

This is a tractrix. It’s actually the same shape as the curve that you get when you drag a pole along a line in the ground. You can imagine this by imagining each card in the spread as a pole at a different instant in time. The pole is being dragged by the lower side of the card, and the upper side is the side being dragged, as shown in this animation.

To make things more interesting, what would the cross-section of these cards look like when rotated about a horizontal axis? Each card would rotate to become a cone, so we would be left with a collection of nested cones that together form the surface of revolution of the tractrix.

This surface is called the pseudosphere, and it is a classic example of constant negative curvature (saddle-shaped curvature). What is cool is that we have built a surface with negative curvature out of many cones, which are intrinsically flat in the sense that they could be built from flat waffles to make waffle cones. This presents itself as a wonderful way to build a pseudosphere from paper: cut out a bunch of identical circles from paper, and roll them up into variously sloped cones, and then stack them from most shallow to steepest.

I calculated the optimum sizes of paper circles and cut them up and created a lovely model of this. It looks like a dunce cap with a very sharp point at the top. In reality, it goes infinitely far up, but I didn’t have enough time to build it that far. I did, however, have enough paper to build the infinite version, because it turns out that the surface area of the pseudosphere is finite despite its infinite extent, which is kind of cool.

The pseudosphere is a series of stacked cones, each with slightly differing slopes. The thing that I found particularly interesting about this model is that the cones have the interesting property that they can all be cut from equally sized disks of paper.

The following video is what this Pseudosphere model looks like in all its glory.

A few years ago, a friend of mine showed me some art by Richard Sweeney, a sculptural artist and designer who built some mathematical structures out of paper.

The first reaction of most people when they see this is confusion. “How could such a structure be built out of paper; it looks solid.” Incredibly, the paper is folded on a curve to give the illusion of solidity. For this reason, as soon as I saw some of his paper forms, I couldn’t help but want to make some myself. Unfortunately, Sweeney had no information on how to build them, so I wanted to rediscover how to build these figures myself. I pulled out my laptop, opened Grapher, wrote down some equations to generate curves for cutting and folding, and printed the result 20 times. After cutting them all out, making the appropriate folds, and gluing it all together, I was pleasantly surprised with my result.  These kind of models look complicated, but actually have quite a bit of symmetry. The icosahedron model, for instance, is made up of twenty “modules” or pieces that fit together. I didn’t have a blog then, so my friend posted the template and instructions on her blog. (We didn’t ask Richard Sweeney for permission, and he didn’t like that, but he allowed the post to stay up if the template is used for non-commercial purposes.)

Each “petal” of the paper model corresponds to a triangular face on the basic icosahedron structure, shown here in green. Every module is identical to every other module. If we can create a paper structure that is symmetrical and has a triangular base, then we can affix many of these onto each face of the icosahedron. Now it is possible to build a structure with icosahedral symmetry, but with much more interesting geometry, so it looks more like a starburst than an icosahedron.

When I first looked at Sweeney’s icosahedron, It struck me as looking more like a dodecahedron than an icosahedron. It can in fact be viewed this way also but requires that the “faces” be reinterpreted to be the five-petaled flowers instead of the triangular protrusions. This ambiguity in interpretation is interesting, and it shows that these two platonic solids are in some sense dual to each other. What is meant by this is that the faces of one polyhedron become the vertices of the other, and edges are drawn between vertices that were previously adjacent faces. We can see the duality between the icosahedron and the dodecahedron in this paper sculpture. In fact, the icosahedral structure of this paper sculpture can be seen best if you were standing inside a giant one, (or if you were very small and standing inside a normal sized one).

Here is a colored version of my final product as an art piece.