My first exploration of polyhedra began with my pen sculptures. I use a lot of pens (It comes with filling up about 80 notebooks over the course of my childhood.) and it kills me to throw them all away. So in middle school I started collecting all the pens in a bag with the plan to someday turn them into art. My vision was to build the platonic solids out of these pens, using the pens as the edges of the polyehdra. Unfortunately, I didn't find a way to justify the time it'd take to build these until a school recycled art project gave me an excuse. So, with duct tape and patience, I constructed the platonic solids out of pens.

It turns out that that takes a lot of pens! (6 from tetrahedron + 12 from cube + 12 from octahedron + 30 from icosahedron + 30 from dodecahedron = 90, in fact. (Notice that duals have the same number of edges!)) More than that, it takes a lot of pens of the same length. If the edges of a given polyhedron vary in length, it won't be regular anymore. I had a lot of pens, but most of them were wildly different lengths. I had to cut a few of the pens down to make them a length that matched other pens, I had to tape the cap of one pen to the body of another, and there were a few cases where I used an ink cartridge on its own rather than a full pen. The triangular solids--the tetrahedron, octahedron, and icosahedron--supported themselves well, but the cube and dodecahedron struggled to stay solid. I had to interweave a few paper straws as internal support.

Unfortunately, my mom made me throw the solids--and the remaining un-polyhedralized pens--away, an understandable but frustrating response to the way they were cluttering my shelf and slowly wilting under their own weight. (That dodecahedron was never going to survive structurally much longer than a day.) But these are still probably one of my favorite art projects ever.

I thought I knew sonobe units backwards and forwards, but then, while preparing to lead an origami class for my church, I stumbled upon a pdf with variations on the sonobe unit and an extensive list of different models you can build with each variation.

My goal is to someday figure out how to construct an origami disdyakis triacontahedron. I don't think any patterns exist, so I'll have to figure one out myself. You'll appreciate the appeal once you see this beauty:

The edges of the disdyakis triacontahedron trace out the mirror planes of the icosahedral symmetry group, which gives it a complex but symmetric structure that I've found delicious since I first encountered the disdyakis triacontahedron in a story I love. (I'll admit, I'm also just generally drawn to Catalan solids.) As far as I know, there are no existing modular origami models of the disdyakis triacontahedron, so I'm going to have to get really good at origami so I can design my own!