First put a carbon atom at each corner and face of a cube. Then put 4 more inside the cube, at the centers of some tetrahedra formed by first ones.

If these 4 extra carbons are the corners of a regular tetrahedron, you get the pattern of carbon atoms in a diamond!


How origami is engineering new technological opportunities ( interview with mechanical engineer Sachiko Ishida of Meiji University on applications of folded structures in engineering, and where origami engineering is headed.

It's here!'s new textbook on category theory will open up the subject to many more people.


A number is 𝘁𝗿𝗮𝗻𝘀𝗰𝗲𝗻𝗱𝗲𝗻𝘁𝗮𝗹 if it's not a root of any polynomial with integer coefficients. Almost every real number is transcendental - but it's often hard to show a specific number is transcendental.

π is transcendental, and so is its partner ϖ.



I have no idea whether it's useful for anything, but a threeld is a pair of fields where the multiplication operation on the inner one forms the multiplication on the outer one. The finite ones have inner order 3 and outer order 2, or inner order a Mersenne prime and outer order the adjacent power of two, but there also exist infinite ones with inner field of characteristic 0 and outer of characteristic 2.

A while ago @davidphys1 asked why nobody had made animations of the shunting yard algorithm with cutesy trains.
There is no surer way to summon me!
I've spent some of my spare time over the bank holidays making exactly that:

Seen on Reddit ...

Q: Why do asteroids always land in craters?

A: That's where they come from. Most people don't realize it, but meteors have much more in common with salmon then they do other heavenly bodies like planets, stars, and moons. Meteorites begin life on earth in craters, then eventually they come back to earth to mate. They seek out the crater where they were born to start the cycle all over again.

I'll stop here. From here, one can get into 2-forms or move to higher dimensions and so on, but for all that, I refer you to the references.

Thanks, as always, for reading.


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Students sometimes wonder why we use quaternions, which are 4D, to represent 3D rotations, when we can use (2D) complex numbers to represent 2D rotations. But that's the wrong way to count dimensions. Complex numbers represent similarity transformations in 2D, which combine 2D rotations (1 degree of freedom) and uniform scalings (another 1 degree of freedom). But 3D rotations have 3 degrees of freedom!

Found in the box of eggs I just bought. Apparently the chicken is calculating a Lorentz contraction.

Claas Voelcker on academic work-life balance:, via

I think we all know that many academics (myself included!) struggle to keep our weekend and evening time free of work-related distractions. Voelcker investigates where this pressure to work comes from (often internally) and suggests that overwork may block creativity; taking time off can make you more productive.

Here's a new illustration for the Wikipedia article on Mrs. Miniver's problem,

The problem asks to arrange two circles so that their intersection (yellow) has the same area as the surrounding parts of the union (blue). It involves solving a transcendental equation, so this seemed like a good time to write a script to find the geometry numerically before switching to a graphics editor to color it in. Source code at


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