July 27, 2016
Topological Crystals (Part 2)
Posted by John Baez
We’re building crystals, like diamonds, purely from topology. Last time I said how: you take a graph $X$ and embed its maximal abelian cover into the vector space $H_1(X,\mathbb{R})$.
Now let me back up and say a bit more about the maximal abelian cover. It’s not nearly as famous as the universal cover, but it’s very nice.
July 23, 2016
Topological Crystals (Part 1)
Posted by John Baez
Over on Azimuth I posted an article about crystals:
- John Baez, Diamonds and triamonds, Azimuth, 11 April 2016.
In the comments on that post, a bunch of us worked on some puzzles connected to ‘topological crystallography’—a subject that blends graph theory, topology and mathematical crystallography. You can learn more about that subject here:
- Tosio Sunada, Crystals that nature might miss creating, Notices of the AMS 55 (2008), 208–215.
I got so interested that I wrote this paper about it, with massive help from Greg Egan:
- John Baez, Topological crystals.
I’ll explain the basic ideas in a series of posts here.