This sculpture (designed by Chaim Goodman-Strauss) has something to do with it!

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My latest puzzle (a belated Christmas present to myself) is the Hanayama Twist. I'm very pleased with it: an elegant symmetric two-piece design, solid feel in the hand, and a solution that is surprisingly complicated but not tediously long. My only complaint is that the solution is very linear, with almost no ways to go wrong if you keep moving on from things you've already done. Anyway here's a map I drew to help me, also used as an example of an implicit graph in my graph algorithms lectures.

A paper-craft model of a chunk of the gyroid, a triply periodic minimal surface.

Think of a number, and keep a running total starting at 0. Each turn, add your number on to the total. Then, if the old total was a multiple of your number, add one to your number. Otherwise, subtract 1.
The game ends when your number is 1.

Which starting numbers eventually get to 1? In the video, it looks like starting at 4 doesn't, but starting at 2 does.

My new sequence,, lists the numbers that eventually reach 1. I'd love to know if there's a pattern!

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Complete classification of tetrahedra whose angles are all rational multoples of \(\pi\):, via

The original paper is "Space vectors forming rational angles",, by Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein

I've got a new sequence in the OEIS. It's to do with a really simple number game I made up. Here's a video explaining the rules

The skew dice set by The Dice Lab is out now at, featuring the most cursed d20 you're ever likely to see!
Full video at

An animation for Craig Kaplan’s art project, with Saul Schleimer. Here we fly through a cohomology fractal that happens to contain a checkerboard. You can explore it at (Open controls, go to the "cool examples" census, and choose "t12047".)

A collection of cohomology fractals for closed manifolds. The four manifolds (in reading order) are m280(1,4), s227(6,1), s400(1,3), and s861(3,1) from the SnapPy census. With David Bachman, Matthias Goerner, and Saul Schleimer.

Our big writeup, "Cohomology fractals, Cannon-Thurston maps, and the geodesic flow", is now up on the arXiv at With David Bachman, Matthias Goerner, and Saul Schleimer. We describe the relationship between cohomology fractals and Cannon-Thurston maps, we give implementation details for our software, and we investigate the limiting behaviour of cohomology fractals, as the "visual radius" increases.

Mechanical Computing Systems Using Only Links and Rotary Joints
Article by Ralph C. Merkle and Robert A. Freitas Jr. and Tad Hogg and Thomas E. Moore and Matthew S. Moses and James Ryley
In collections: Basically computer science, Things to make and do, Unusual computers
A new paradigm for mechanical computing is demonstrated that requires only two basic parts, links and rotary joints. These basic parts...
Entry: read.somethingorotherwhatever.

The dLX (pronounced "d-Lex", as in "lexicon"), is a new 60-sided, alphabetical die from The Dice Lab. Sixty is enough for us to get a letter distribution that is close to the distribution in the English language, so they can be used for word search games!

It may look like we are descending to land on an alien planet, but this is actually the new "hyperideal" view in our cohomology fractals web app. With Saul Schleimer, David Bachman, and Matthias Goerner.

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