Laird Robert Hocking

Statement

As a kid, I grew up exploring coastal British Columbia, sometimes bringing home treasure that I found. I would sometimes feel an intense need to find out what the view is like from a mountain visible in the distance. Similarly, as an adult I find myself intensely curious about, for example, "What do 3D cross-sections of 4D Menger sponges look like?". In both cases, an answer exists - I just have to go out and find it (either by climbing the mountain, or by doing the math). In this spirit, I see mathematical art less as a creative process and more one of exploration and discovery. Just like when I was a kid, I rarely find anything interesting. However, I occasionally stumble up a treasure which I want to take home (with a 3D printer).

Artworks

As a mathematical artist, my main goal is to make abstract ideas tangible, ideally as physical objects you can see and touch.  Certain kinds of beautiful mathematical objects, in this case 3D Kleinian limit sets, are challenging because 3D printing them is infeasible.  Therefore, this project uses a different artistic medium, a digital hologram, which creates the illusion of a 3D object floating in space by means of a diffraction grating that scatters light exactly as if the object were there. To create one, I needed to render my 3D limit set from 69120 different angles, which took my computer 800+ hours and over a month.  Holograms are notoriously difficult to photograph, but I have done my best with two photos from two different angles.

Inspired by the hexagram holes in the Menger slice, I wanted to make a fractal dodecahedron with pentagram shaped holes.  As the mathematics is less perfect in this case, I ended up settling on three different designs, each with their own pros and cons.  In one case, the tips of the pentagrams are chopped off.  In another, those tips are extended, but at the cost of increasing the amount of "filler metal" used (each fractal is made up of smaller copies of itself plus filler).  In the third, the pentagrams are narrowed instead of extended, which fixes the problems of chopping but results in a fractal that breaks down after two recursion steps.  Perhaps a fourth design better than all three is still out there, waiting to be discovered?