If you roll a circle inside a circle 3 times as big, you get a deltoid
. If you roll a circle inside one 4 times as big, you get an astroid
. But here's something cool: you can roll a deltoid inside an astroid, and it fits perfectly!
This movie was made by Greg Egan. The deltoid looks like a curved triangle with sharp corners. The astroid looks like a curved diamond with sharp corners. Note how all 3 corners of the deltoid always touch the astroid.
Why does this happen? You can see an explanation on my Visual Insight
It's related to some things that physicists like: SU(3) and SU(4).SU(n)
is the group of n × n unitary matrices with determinant 1. Physicists like Pauli and Heisenberg got interested in SU(2) when they realized it describes the rotational symmetries of an electron. You need it to understand the spin
of an electron. Later, Gell-Mann got interested in SU(3) because he thought it described the symmetries of quarks. He won a Nobel prize for predicting a new particle based on this theory, which was then discovered. We now know that SU(3) does
describe symmetries of quarks, but not in the way he thought.
It turns out that quarks come in 3 colors
- not real colors, but jokingly called red
. Similarly, electrons come in 2 different spin states, called up
. Matrices in SU(3) can change the color of a quark, just as matrices in SU(2) can switch an electron's spin from up to down, or some mixture of up and down.
SU(4) would be important in physics if quarks came in 4 colors. In fact there's a theory saying they do, with electrons and neutrinos being examples of quarks in their 4th color state! This is called the Pati-Salam theory
. It's lots of fun, because it unifies leptons (particle like electrons and neutrinos) and quarks. There's even a chance that it's true. But it's not very popular these days, because it has some problems: it predicts that protons decay, which we haven't seen happen yet.
Anyway, the math of SU(3) and SU(4) is perfectly well-understood regardless of the physics. And here's the cool part:
If you take a matrix in SU(3) and add up its diagonal entries, you can get any number in the complex plane that lies inside a deltoid
. If you take a matrix in SU(4) and add up its diagonal entries, you can get any number in the complex plane that lies inside an astroid
. And using how SU(3) fits inside SU(4), you can show that a deltoid rolls snugly inside an astroid!
The details are in my blog article. And the pattern continues! For example, an astroid rolls snugly inside a 5-pointed shape, thanks to how SU(4) sits inside SU(5). Here's a movie of that, again made by Egan:http://gregegan.customer.netspace.net.au/images/astroid5.gif