Exploring Klein's Quartic: Need Help!

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In summary, the conversation discusses difficulties understanding the geometrical construction of Klein's quartic, specifically how to obtain a 14-gon with 336 triangles and the actions of the group PSL(2,7) on this figure. The individual also requests help and mentions a helpful web page by John Baez.
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mery2
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Hi all!

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

Starting from the fundamental region [itex]R=\{z\in \mathbb{H}| |z|>1,-\frac{1}{2}\leq Re(z) \leq \frac{1}{2}\}[/itex], 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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  • #2
mery2 said:
Hi all!

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

Starting from the fundamental region [itex]R=\{z\in \mathbb{H}| |z|>1,-\frac{1}{2}\leq Re(z) \leq \frac{1}{2}\}[/itex], 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
 
Last edited:

1. What is Klein's quartic?

Klein's quartic is a mathematical object discovered by German mathematician Felix Klein in the late 19th century. It is a Riemann surface, which can be thought of as a 2-dimensional curved space. It has the interesting property of being self-dual, meaning that it looks the same when viewed from different angles.

2. Why is Klein's quartic important?

Klein's quartic has been an important object of study in various areas of mathematics, including geometry, topology, and number theory. It has also been used in physics, specifically in string theory and quantum field theory. It has also been studied for its aesthetic properties, as it has symmetries that make it visually appealing.

3. How many points does Klein's quartic have?

Klein's quartic has 24 points, also known as the Klein's quartic curve. These points are the solutions to a particular polynomial equation in three variables. Each point has a unique set of coordinates on the surface.

4. Can Klein's quartic be visualized?

Yes, Klein's quartic can be visualized using various techniques, such as computer-generated images or physical models. However, due to its self-dual nature, it is difficult to represent in a traditional 2-dimensional space. It is often depicted as a tiling pattern or a projection onto a 3-dimensional space.

5. What are some applications of Klein's quartic?

Klein's quartic has applications in various areas of mathematics and physics, including algebraic geometry, topology, and string theory. It has also been used in cryptography, specifically in the development of codes and ciphers. Additionally, the symmetries of Klein's quartic have inspired artwork and design in fields such as architecture and fashion.

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