# Klein's quartic

## Main Question or Discussion Point

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...?

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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...?