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File:Symmetries of the anharmonic group.png

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English: The anharmonic group of permutations of the cross-ratio can be visualized as the rotation group of the trigonal dihedron, which is isomorphic to the dihedral group of the triangle D3.

Considering {0, 1, ∞} as the vertices of a triangle, and the anharmonic group as its stabilizer, the fixed points of the 2-cycles are the points {−1, 1/2, 2}, which correspond to the midpoints of the edges (and the 2-cycles as rotation about that edge), and the fixed points of the 3-cycles are , which correspond to the top and bottom poles (centers of the sides; the 3-cycles correspond to rotation through their axis), and are interchanged by the 2-cycles.

In this diagram, is the point at infinity.
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Author Jacob Rus

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The anharmonic group of permutations of the cross-ratio can be visualized as the rotation group of the trigonal dihedron, D3

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14 January 2023

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