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  I worked with George Bell and made a puzzle: Dodecahedron Assembly Kit. There are 3 types of pieces. The challenge is to use some of them to assemble into a dodecahedron. The whole set can form two dodecahedra in at least 2 ways. There are even more ways to assemble some pieces into one dodecahedron with the leftover can't assemble together. Some of the assemblies have 2-fold, 3-fold and 5-fold symmetries, and some don't have any symmetry. Every assembly has its unique challenge...

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  I made Lights Out 4D: the classic puzzle game, but on stereographic projections of 4D polytopes. Three shapes to solve: 16-cell → 24-cell → 600-cell. For these polytopes, edges form great circles (rings). Players click a vertex to toggle the state of the rings passing through it. The objective is to turn off all rings. You can change viewpoints in 3D and 4D. Works on phone & desktop. Link: https://www.nan.ma/lights_out_4d/

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  My favorite Pi Day thought this year comes from a question I saw online: Is π a complex number? Of course π is a real number, and every real number is also a complex number. Still, almost nobody casually says “π is a complex number.” In the same spirit, I wouldn’t normally say π is a quaternion either. It’s a funny situation: math is supposed to be as rigorous as possible, yet in ordinary language we hesitate to say some perfectly correct things. I’m happy to write \pi \in \mathbb{C}, but the...

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  I built a paper model of a great dodecahedron. The net consists of 12 pentagonal "flower" shapes, each in a different color. What makes this particular net interesting is that it also serves as a net for a regular dodecahedron: the same flat pattern can be folded into either shape. This dual property is quite rare: most dodecahedron nets do not work as great dodecahedron nets. With an arbitrary net, two faces will typically end up in the same position while another face is miss...

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  Follow-up to my paper great dodecahedron post https://mathstodon.xyz/@nanma80/116303406475501952 I enumerated all common nets of the dodecahedron and the great dodecahedron. They are nets with 12 pentagonal faces that fold into both shapes, with faces interpenetrating when folded into the great dodecahedron. With most nets, some faces collide (land in the same position) and not all 12 face positions are covered. Out of 43,380 distinct nets (up to the full icosahedral symmetry), exactly 74 fold...

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  Various stages of the motions

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  We can fold a net into a dodecahedron as in a paper model. If faces can intersect, we can further fold it to a new shape. 12 original faces overlap to get only 6 unique faces, with one as base and 5 side faces. It can be seen as a stellated pentagonal pyramid. The stellated pentagonal pyramid isn't a real polyhedron because some edges belong to only one face. In a real polyhedron, each edge must belong to exactly two faces.

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  I'm still having fun with the yellow Cairo tiles from my previous post. This time, I mixed them with the white hexagonal tiles from an earlier model to create a flowery pattern. The key observation is that six Cairo tiles can form a ring with a regular hexagon at the center. If we fill that central hole with a white hexagonal tile, the ring looks like a yellow-and-white hexagonal flower. If we leave the hole open, it becomes a large hollow hexagon. Once the pieces are grouped into thes...