Klein’s quartic curve is a surface of genus 3: a 3-holed torus, shown here embedded in R^3. But this gives a distorted view of its geometry; in the true surface, all 336 triangles are congruent, and there is a symmetry of the whole surface that maps any of them into any other.pic.twitter.com/7yYHqnWSgK
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Here's an article by Moore and Schlafly "On Equivariant Isometric Embeddings" which discusses equivariant isometric embeddings: https://link.springer.com/article/10.1007/BF01159954 … Interesting that there's no group-independent bound on the embedding dimension!
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There’s an easy way to get a (non-isometric) equivariant embedding with all 336 symmetries. Take the 24 vertices of the 56-triangle tiling of KQC, map them to the std basis in R^24, and map the hyperbolic triangles to Euclidean triangles. This surface will lie in copy of R^23.
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