One, Two, Four

At the MSRI in Berkeley, there is a marble sculpture by Helaman Ferguson showing Klein’s quartic surface.Kleinquartic1

This is a Riemann surface of genus 3 with 168 automorphisms. Our Euclidean brains have a hard time seeing all these. Let’s start with an automorphism of order 7, and a tiling of the plane by π/7 triangles:

Heptatile

Fourteen of them fit around a common vertex (at the center of our hyperbolic universe), and the black geodesic indicates how to identify edges of the green-yellow 14-gon (repeat the pattern by 2π/7 rotations). Euler will tell you that the identification space has genus 3. A little miracle is that these π/3 triangles fit nicely into a tiling by π/3 heptagons. This becomes evident like so:

Klein237

The geodesic we used to indicate the 14-gon identification pattern becomes a geodesic in the heptagon tiling that passes through edge midpoints of eight consecutive heptagons, and all such geodesics will be closed on the identification space. This allows to define this surface also as an identification space of 24 heptagons (using the same geodesics). As this description is intrinsic to the heptagon tiling, it is invariant under all symmetries of that tiling, which include rotations of order 2 and order 3, in addition to the order 7 rotation.

Kleinflat 01

Why is this surface called a quartic? Replacing the hyperbolic π/7 triangles with Euclidean (1,2,4)π/7 triangles in three different ways and keeping the identifications, we obtain three different translation structures on the Klein quartic, which define a basis of holomorphic 1-forms. Playing with their divisors show that these 1-forms satisfy the equation x³y+y³z+z³x, showing that the canonical curve of Klein’s surface is a quartic curve in the complex projective plane.

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