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

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SUMMARY

The geometrical construction of Klein's quartic involves a fundamental region R=\{z\in \mathbb{H}| |z|>1,-\frac{1}{2}\leq Re(z) \leq \frac{1}{2}\}, which leads to the creation of a 14-gon composed of 336 triangles. The group PSL(2,7) acts on this figure, with specific edge identifications such as 1-6, 3-8, and 5-10. John Baez provides a comprehensive outline of this construction, emphasizing the relationship between heptagons and triangles in the tiling of Klein's quartic.

PREREQUISITES
  • Understanding of complex analysis, specifically the upper half-plane model.
  • Familiarity with group theory, particularly the properties of PSL(2,7).
  • Knowledge of geometric constructions involving polygons and triangulations.
  • Basic comprehension of tiling theory and duality in geometry.
NEXT STEPS
  • Study John Baez's webpage on Klein's quartic for detailed construction insights.
  • Explore the properties and applications of PSL(2,7) in geometric contexts.
  • Research the relationship between heptagons and triangles in tiling theory.
  • Investigate the implications of edge identifications in polygonal constructions.
USEFUL FOR

Mathematicians, geometry enthusiasts, and students studying complex analysis and group theory who are interested in advanced geometric constructions and their applications.

mery2
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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...?

Can I ask for your help, please?
Thank you in advance!
 
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mery2 said:
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
 
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