This drawing is titled "Drawn Entirely With Straight Lines", and it has some interesting geometric properties. I’ll explain how it works.
First, we can mentally break the hexagon shape into smaller modules:
Each equilateral triangle region is identical, and they are reflections of each other. If you placed this hexagon inside of a kaleidoscope, this pattern would repeat infinitely in all directions.
The imaginary curves that are visible in the picture are closely related to the mice problem, which is a special kind of pursuit problem (like what we saw in my tractrix post). The mice problem goes like this: Three mice are initially sitting at the corners of an equilateral triangle. All at once, each of the mice begin crawling with equal speed directly toward the mouse on their right. What is the path of each mouse?
This is an animation of the path of the rodents (not made by me):
Suppose that the rats periodically and simultaneously leave droppings to mark their path. Again, each triple of simultaneous droppings forms an equilateral triangle. What would this look like? I used Mathematica’s Graphics function to illustrate it:
In this animation, the rodents leave droppings more frequently as their collision becomes more imminent (perhaps in eager anticipation of their meeting). The rodents always form an equilateral triangle, which means that the path that they take is also symmetrical — it is a simultaneous rotation and scaling (shrinking) of the triangles. Here is another animation of this effect:
It is precisely the rotation/scale symmetry that allows this gif to loop seamlessly.
But why do the lines curve like that? The angle between the direction a rat is aimed relative to the center of the triangle remains fixed throughout, and the resulting kind of curve is a logarithmic spiral. If you really use your imagination, as a regular screw can sink into wood, a screw with this logarithmic shape could be theoretically used to enlarge/scale something.
Anyway, here is a quick sketch with pencil and paper I made using the mouse algorithm with step size remaining constant (constant dropping rate):
I also did a similar diagram in Mathematica, again with the triangles more dense toward the center to emphasize the symmetry. That is, if each triangle were rotated and scaled the right amount, it would look identical, except with either an extra or a missing outer triangle.
There is a sculpture in San Francisco that looks like a three-dimensional icosahedral illustration of the mice problem (it’s called Icosaspirale, by Charles Perry). This time, each triangular module hides a similar scale/rotation symmetry, except now in three dimensions, with the center of dilation not lying on the plane of the largest triangle, but instead somewhere between the center of the triangle and the center of the icosahedron. It is really remarkable what kind of astounding visual effects can be generated by imposing various symmetries!
Mathematical equations are always used to make predictions and carry out proofs. I wanted to break this trend and explore the ways that equations could be used qualitatively to describe art — in particular, how they can be used to paint Easter Eggs. Using Mathematica, I created a three-dimensional egg shape via the parameterization via a perturbation of a sphere (c=0.2=,b=1.65):
This surface has similar geometry to an egg, but is lackluster. At this point, it still has meshlines and a generic default color assigned by Mathematica. The eggs need some more spice. I decided I wanted to color them like Easter Eggs, but the constraint I imposed on myself was that the colored patterns would need to be created directly from mathematical functions inside of Mathematica. Each function would go from the surface of the surface of the sphere into the real numbers, and therefore could be thought of as a simple function of two variables, u and v. The output of this function could then be mapped to a color to represent the desired color at that point.
The first mathematical pattern that immediately came to mind for me was the SphericalHarmonicY function. This function is used for solving Laplace’s equation in spherical coordinates, so it already has a natural setting in our egg shape which has the same basic structure as a sphere. I then chose a color scheme that fit well with the pattern I had created. Here is an example of my Spherical Harmonics egg painted with the gradient “BlueGreenYellow”:
Thankfully, there were no problems matching up edges on the rectangle [0,2π]x[0,π], because the Spherical Harmonics are made especially for spheres. At this point, I couldn’t help but try changing parameters on the eggs to obtain different patterns. For instance, one such parameter represents the number of blobs of color going from the top to bottom of the egg. The other parameter represents how many blobs go all the way around the equator. By changing these parameters, you can obtain very different looking eggs, especially when you make one parameter much larger than the other to create “slits” of color. You can see an orange version of these Spherical Harmonics in the bottom picture in the orange-colored egg on the very left of the image. You can see that I cranked up the number of blobs of color I required to fit from north pole to south pole in that one.
At this point, I decided I needed to get a little bit more creative. Using predefined functions like SphericalHarmonicY was fun and all, but it also left me somewhat unsatiated. I began playing around with spirals, and here is one such example of an array of spirals with alternating handedness all over the sphere.
The first think I noticed about the egg is the spirals, but if you look directly down on this model from the north or the south pole, you will see a four-petaled flower. What I find interesting about this activity is that it requires not only artistic creativity, but also a familiarity with the behavior of mathematical functions so that one can define an egg that resembles the image one had in ones mind before beginning the project. However, unlike this visually attractive egg, the mathematical coloring equation is no where near as elegant. Ignoring various rescaling done to account for the desired input range to specify the colors, the mathematical equation that I used to generate this pattern was:
All together, I made eleven Mathematically colored eggs that I considered worthy of keeping. The most difficult one was probably the Patriotic Star Egg shown in the bottom right corner. It required several heavy-duty functions working together to make the repeated star print like Floor, Mod, ArcTan, Abs, and Max. (I won’t do you the disservice of posting the entire function used.) Probably my favorite eggs are the two African Zig-Zag Eggs.
I went to a Mexican restaurant and saw these cool lamps in the shape of polyhedra (albeit not platonic solids). The shape is some kind of stellated rhombicuboctahedron, although it is not a “true” one, for two reasons: first, the stellation does not occur by simply extending faces to a point (the point is formed farther than then intersection of faces), and second, only the square shaped faces are stellated on the rhombicuboctahedron.
I really liked them, so I located a version for sale and gave one to my mother as a gift. The website calls them Moravian stars. Here is the lamp on our table:
My friend has found Stellated Dodecahedron lamps in Santa Barbara:
Unfortunately it’s hard to find a similar lamp in icosahedron form, but there’s an IKEA lamp that is based on the icosahedron shape.
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.