Nan Ma

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.

The 3 types of pieces are tightly related. The smallest pieces (bottom row) are the basic shape. I call them singles. The middle two rows are two singles connected together (doubles). The largest pieces are five singles connected together (quintuples). Each dodecahedron is formed by 20 single units.

I analyzed the ways that the pieces can move. I'm visualizing them in this webpage:
https://observablehq.com/d/008311b8a651406d
There are 8 distinct motions. All the allowed motions are convex combinations of them.

Follow-up to my paper great dodecahedron post
@nanma80@mathstodon.xyz

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 into both shapes. 26 of the 74 have a 2-fold rotational symmetry; the other 48 are asymmetric. Symmetric nets are heavily overrepresented: only 0.8% of all dodecahedron nets have any symmetry, but among common nets it's 35%. The property of common nets somehow greatly favors symmetric nets.

The most familiar dodecahedron net, two "flowers" of 6 faces each, is not a common net. But some variations with two clusters are. Image 1 is the net I used for my paper model. Other images are a few more of the 74.

All 74 nets:
https://github.com/nanma80/star-polytope/tree/master/output/common_nets_dodecahedron_great_dodecahedron

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/

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 missing entirely. Folding it into the great dodecahedron is the challenging part: the faces penetrate and interlace with one another, requiring the paper to be bent and woven together.

The first part of this video: @nanma80@mathstodon.xyz
shows how this net folds into a dodecahedron and then into a great dodecahedron, when the faces are allowed to pass through each other. This paper model is my attempt to bring the simulation into the physical world.

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 English sentence “π is a complex number” somehow feels a little strange.