Boxes don’t have to be boring
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.