28 Oct

I’ve used a lot of different materials to build SierpiĆski Tetrahedra over the years, including mailing tubes and hula hoops, among other possibilities.

You can see a couple of these previous efforts in the left and right photos below.

So when I discovered that the Lakeshore Learning Ultimate Fort Builder construction toy allowed rods to be connected at all of the angles necessary to construct arbitrary portions of an octahedral-tetrahedral lattice, I knew that I had no choice but to eventually seek yet again to scale this tower of three-dimensional fractality.

The opportunity came in the fall of 2022, when my colleague Prof. Alissa Crans put me in touch with Profs. Derek Smith and Ethan Berkove, who were interested in holding a collaborative mathematical construction outside the math department’s building on the Lafayette College campus. After reviewing a number of proposals, they (to my delight) selected a Sierpinski tetrahedron constructed from the Fort Builder toy. They felt this structure would tie in to a number of classes being held that semester, including one on combinatorics — just how many rods and joints are there in that pyramid, anyway?

This post will be fleshed out as time permits, but for now suffice it to say that the construction — quickly dubbed Fort Sierpinski — came together very satisfyingly (more below).

The final structure came in at just a few inches shy of twenty feet tall, making it the largest Sierpinski tetrahedron I’ve been involved in yet, as well as the highest-order approximation: there are five generations of tetrahedra, from the smallest unit tetrahedra to the construction as a whole.

If you look closely, you will find just a few connections where the ability of the hub-rod sockets to take tension trying to pull them apart had to be augmented with a little strategically-placed tape (as hinted by the items on the table in the foreground). Other than this kind of patching (after all, kids do have to be able to take their creations apart by pulling out the rods, and this giant building is putting a lot of tension on certain junctions), the Fort Builder toy turned out to be quite up to this gargantuan task.