File:Klein quartic with closed geodesics.svg

The fundamental domain of the Klein quartic has 336 right triangles.
They have three types of sides, hereafter called segments:

• short leg (S) between 3-fold and 4-fold points
• long leg (L) between 7-fold and 4-fold points
• hypotenuse (H) between 3-fold and 7-fold points

The 504 segments form 28 closed geodesics, each consisting of 18 segments of these types:
SHL LHS SHL LHS SHL LHS

They are marked with 7 colors and 4 line styles.
Geodesics with less dominant line styles appear more separated in the 14-gon:

• Thick are cut in 2 parts: LLHSSHLLHSSH (12), LLHSSH (6)   (These are the mirror axis and the 14-gon edge, which appears twice.)
• Thin are cut in 3 parts: LLHSSHLL (8), HSSHL (5), LHSSH (5)
• Dashed are cut in 4 parts: HSSHLLH (7), SS (2), HLLHSSH (7), LL (2)
• Dotted are cut in 5 parts: SSHLLHSS (8), HLL (3), HS (2), SH (2), LLH (3)

The geodesics correspond to the vertices of the Coxeter graph.
42 pairs of geodesics have two rectangular crossings (each in the middle of one of the 84 rhombi).
These pairs correspond to the edges of the Coxeter graph.

Own work

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See below.