Nan Ma

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 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.

@liuyao@mathstodon.xyz @divbyzero@mathstodon.xyz I was replying to Dave yesterday on Facebook as well. I don’t think the pencils will stay interlocked just by themselves. We need to use rubber bands or some structures to hold them together. Although we can stack pencils like Jenga to build a tower, it’s less symmetrical as the construction here.

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/

Close-up pictures of the ring of 6 Cairo tiles + 1 hexagon.

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 these larger hexagons, they can tile the plane.

Links to the previous posts for these two tiles:
@nanma80@mathstodon.xyz
@nanma80@mathstodon.xyz

@csk@mathstodon.xyz @henryseg@mathstodon.xyz @hallunke23@troet.cafe the word Swastika may refer to two chiral versions, left and right rotating. The Nazi symbol only has one version. When I’m talking about mirroring the symbol to avoid Nazi, it is confusing to use the word Swastika, so I didn’t use it.

@henryseg@mathstodon.xyz @hallunke23@troet.cafe in my previous work (Splinter-12) I also made sure it’s a mirrored nazi symbol on each face. It turns out after expansion, the pattern becomes a mirrored Chase Bank symbol. Chase Bank and Nazi symbols are turning in the same direction.

@hallunke23@troet.cafe well, that’s part of the mechanism. The best I could do is to let it spiral in the opposite direction as the nazi symbol.

Various stages of the motions

I built a new 2D coordinate-motion model based on the Cairo tiling, a periodic tiling of the plane. All the identical pieces move in and out together.

There are two independent expanding motions, rotated 90° from each other. Each one opens only half of the triangular holes. You can also tell them apart by watching how the central square expands.

If you combine the two motions, all the holes open and the pieces move uniformly.

Since each piece is an irregular pentagon, assembly turned out to be trickier than I expected. If I'm not careful, I end up connecting the wrong sides.

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.

@rasjor@mathstodon.xyz thanks again. The rendering part is relatively straightforward. If you want to hear about the extra algorithm, I enumerated all the nets of the dodecahedron, and programmatically checked which nets will cover all the faces of the great dodecahedron by “overfolding”. Only about 0.1% of the dodecahedron nets cover all faces of the great dodecahedron. Those are rare cases, and the common nets won’t work. Interestingly, 100% of the icosahedral nets can overfold into the great icosahedron covering all the faces.

@rasjor@mathstodon.xyz Thanks a lot! For these folding animations, I wrote a custom algorithm to calculate the rotation of each polygon to form the 3D shape, then render the 3D shapes. At the end these are just a bunch of 3D rotations around lines.

If we start from a different net, we can get the great dodecahedron where all 12 faces are distinct. The stellated pentagonal pyramid contains half (6) of the 12 faces of the great dodecahedron. If we start from others net of the dodecahedron, we may get a shape with between 6 and 12 faces.

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.