You can think of each pentagram here as a pentagon that's been mapped into space in a very distorted way, with a "branch point of order 2" at its center. What does that mean? (2/n) pic.twitter.com/fbdVLn496o

Stand at the center of a pentagon! Measure the angle you see between two corners that are connected by an edge. You get 2π/5. Now stand at the center of a pentagram. Measure the angle you see between two corners that are connected by an edge. You get 4π/5. Twice as big! (3/n)

So, to map a pentagon into space in a way that makes it look like a pentagram, you need to wrap it twice around its central point. That's what a "branch point of order 2" is all about: en.wikipedia.org/wiki/Branch_po… (4/n)

That's the cool way to think of this shape. It's a surface made of 12 pentagons, each wrapped twice around its center, 5 meeting at each sharp corner. If you use Euler's formula V - E + F = 2 - 2g to count its number of holes - its "genus" g - you'll see it has 4 holes. (5/n) pic.twitter.com/vb9ifpX78t

It's actually a Riemann surface, the most symmetrical Riemann surface with 4 holes! We're seeing it as a branched cover of the sphere. But you can also build it by taking a tiling of the hyperbolic plane by pentagons, and modding out by a certain group action. (6/n) pic.twitter.com/4sZVtzP56s

Most of this stuff - and more - was discovered by Felix Klein in 1877. You can read details in this blog post of mine: blogs.ams.org/visualinsight/… (7/n, n = 7) pic.twitter.com/0p97aKGfa3

Amazing. It should be like a football (soccer ball).