Aaron Fenyes
Drawing pictures of the two-dimensional surfaces I study isn't just a treat, or a way to communicate; it's a research tool I couldn't do without. I use hand-drawn sketches to get a feel for how geometric objects behave—to see which ways they can move and bend and still look true to life. I use computer drawings to explore unfamiliar objects I can't see well enough yet to sketch, and to see familiar objects with more clarity than my intuition can provide.
The drawings in my first submission come from software I wrote to test the results I proved in my Ph.D. thesis. I hope they'll show you a glimpse of what I find most beautiful about my field.
A pineapple's florets fall into three families of spirals. Urban legend has it that if you tie a string around a pineapple and count the spirals from each family passing through it, you'll typically get three successive Fibonacci numbers. Many plants' florets follow the same pattern, and many models have been proposed over the years to explain why (see Mughal and Weaire, 2017, for a recent one). I bought this pineapple from a Waterloo, Ontario Valu-Mart in 2010, intending to make it a poster child for the "three Fibonacci numbers" rule. Unfortunately, between photographing it and eating it, I forgot to count the spirals. Now I'll never know whether it followed the rule I'm using its image to illustrate.
This poster shows two pictures of the same hyperbolic surface. The surface is divided into four colored triangles, all the same shape and size. To make the picture above, I unrolled the surface onto the hyperbolic plane, where the four triangles form a tessellation. Below, I flattened the surface, collapsing the triangles to tripods.* Then I cut it into a polygon, with the tripods' centers at the vertices. The tripods' legs spiral around the surface, filling it as densely as the rational numbers fill the real line.
The poster was handmade by a Toronto print shop. Natural printing variations evoke the interleaving of the triangles, less perceptible in the more precise digital catalog print.
* See Gupta's "Asymptoticity of grafting...."