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In the world of geometry, the small stellated dodecahedron is a superstar! It's a star made of stars. It has 12 pentagrams as faces. But it's also the most symmetrical Riemann surface with 4 holes. Huh? Let me explain. (1/n)pic.twitter.com/ZnzN0qMEx3
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: https://en.wikipedia.org/wiki/Branch_point … (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: https://blogs.ams.org/visualinsight/2016/06/15/small-stellated-dodecahedron/ … (7/n, n = 7)pic.twitter.com/0p97aKGfa3
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