How Can I Understand the Geometrical Construction of Klein's Quartic?

[mery2]

Hi all!

I have some problems understanding the geometrical construction of Klein's quartic.

Starting from the fundamental region $R=\{z\in \mathbb{H}| |z|>1,-\frac{1}{2}\leq Re(z) \leq \frac{1}{2}\}$, how can I obtain a 14-gon with 336 triangles?
Moreover, how does the group PSL(2,7) act on this figure? Why the edges' identifications are exactly 1-6, 3-8, 5-10...?

Can I ask for your help, please?
Thank you in advance!

 

[Anti-Crackpot]

Hi all!

I have some problems understanding the geometrical construction of Klein's quartic.

Starting from the fundamental region $R=\{z\in \mathbb{H}| |z|>1,-\frac{1}{2}\leq Re(z) \leq \frac{1}{2}\}$, how can I obtain a 14-gon with 336 triangles?
Moreover, how does the group PSL(2,7) act on this figure? Why the edges' identifications are exactly 1-6, 3-8, 5-10...?

Can I ask for your help, please?
Thank you in advance!

Hopefully, I am not misremembering anything here, but Klein's quartic is composed of 24 heptagons, each of which can be decomposed into 14 triangles. 24 * 14 = 336. John Baez has an amazing web page outlining the construction. That might be a good place to start to answer your questions. Note that Baez also explains how Klein's Quartic can (dually) be tiled by Triangles instead of Heptagons, but he focuses on the Heptagonal Construction.

- AC