Category Archives: Knots

Hair Tie 120-cell

Image by Lucas Garron

About two years ago, I decided to make a 120-cell out of hair bands. I had just come back from Gathering for Gardner 10, where I had helped my friend Zach Abel build a cool, giant rubber band sculpture. This sculpture was fantastic and I loved how boingy it was. I was incredibly inspired by his idea and technique of using elastic bands to create geometric sculpture, and I wanted to try making my own.

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Shortly thereafter, Vi Hart asked me what my favorite polychoron was, a question to which I immediately answered “the 120-cell“.

See, my favorite number is five (a long-standing fact that first emerged when I was 5 years old), and, by extension, my favorite polygon is the five-sided pentagon. It naturally follows that my favorite polyhedron is the regular dodecahedron comprised of 12 regular pentagons. The 120-cell, made out of 120 regular dodecahedra, was basically a shoo-in for my favorite 4-polytope.

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Thus, when I discovered CVS selling colorful hair bands by the hundred, it seemed obvious that I needed to make a giant, bouncy (projection of a) 120-cell out of them. Hair bands seemed like a natural thing to use for a 120-cell. Their natural stretch means that you don’t need to get the lengths exact (just close), they come in all kinds of colors, and they have a truly delightful sproing to them which encourages people to really interact with the final sculpture.

I immediately purchased 600 hair bands (buying out the CVS in question), and got started. I particularly like the way that the Schlegal diagram of the 120-cell shows all of the edges and vertices and has a regular dodecahedron in the very center and one on the outside, so I knew that my 120-cell was going to use that projection and have 1200 edges, so this seemed like a good start. I assumed that I would be able to return to the store once I’d used up my hair ties and buy some more. Unfortunately, what I hadn’t account for was that CVS doesn’t actually restock that quickly. Moreover, since only the innermost edges of the sculpture were a single band length, and I was using a color scheme to keep track of my place in the sculpture, I was going to need way more than 1200 hair ties.

In desperation, I tried hair tie shopping at a different local store, but they didn’t have the same hair bands. I ended up buying out 5 or 6 CVS all across the Bay Area to finally end up with enough hair bands for my sculpture.

With the structure completed, I ran into a different issue. The sculpture was going to be huge and was going to need to be stretched out from multiple corners. It was probably going to fill a whole room, and I didn’t actually have a spare room handy. Fortunately, my boyfriend had a small extra bedroom in his apartment, and, one weekend we took a bunch of 3M command strips and attached it above the bed, which was awesome, but also a bit strange. This picture was carefully cropped to hide the bed.


Not long thereafter we ended up taking the 120-cell down. Above one’s bed just isn’t a good permanent location, and the 120-cell languished in a bag for over a year before I had another opportunity to put it up in a small room in the CDG office.


Did I mention that the 120-cell is rather large?

Check out this Youtube video to learn more about the hair tie 120-cell.



Zip Tie Tangle

Zip ties are a great way to tie things together, from cables to people. They are extremely difficult to take apart non-destructively once they have been “zipped”, so you have to be careful when trying to build something out of them, but having a built in connection mechanism makes building things “easy”.

Which is to say that it’s easy to build things as long as you want to make long strips or rings of zip tie, but there is no real way to “branch” your rings. Pentagon? easy. Tetrahedron? hrm…

Fortunately, we don’t need nodes of degree greater than 2 to make cool structures. By interleaving rings of zip ties, we can make compounds of regular polygons. These structures are referred to as regular polylinks or orderly tangles.


Orderly tangles are a nice way to illustrate symmetric colorings. I made this orderly tangle of four zip tie triangles out of four different colors of zip tie. The resulting structure has octahedral symmetry. If you think of the zip ties as representing edges, and the spaces between them as faces, the resulting structure in the middle can be thought of as a woven cuboctahedron.

Since the zip tie triangles have a chirality to them, the final sculpture also has some chirality. Keeping the chirality straight during construction may be the most difficult part of making this piece.


Many other tangles are possible. A tangle of six pentagons is tricky (and I don’t have that many colors of zip tie), but would demonstrate a similar symmetric coloring of the icosahedral group. Let me know if you try making one!

Novel decorative knots

are particularly enjoyable to tie. There is something really enjoyable about making a design on paper or in my head come to life. The pan-chang-esque knot with the hexagonal grid is one of my favorites as I really like that grid pattern and was surprised that nobody seems to have ever used it in a large-scale knot. I’m also fond of the turk’s head-ish knot with unusual the “3 in 5” braid pattern.


Sierpinski Triangles

are so much fun to make that they have shown up in this blog before! Here are a couple fun ones that I have made recently. The first one gives you a sense of just how many strawberries I ate on my last strawberry picking trip. The second one was made with thread crochet.



can be used to make very attractive choker necklaces. The bottom three here are choker length. My personal favorite is what I refer to as a “3 in 5” braid as it is a five strand braid that alternates between three and five strands. These braids all have the property that they were braided out of a single length of cord that was looped to form the number of strands in the braid and then braided normally downward. This is possible with all odd stranded standard braids, but no even stranded standard braids with greater than four strands. A second strand is added for color interest, as well as a loop and knot system for clasping (out of a 3-strand braid and a standard knife lanyard or Chinese button knot.