Klein Quartic - Tiling

Tiling

The Klein quartic admits tilings connected with the symmetry group (a "regular map"), and these are used in understanding the symmetry group, dating back to Klein's original paper. Given a fundamental domain for the group action (for the full, orientation-reversing symmetry group, a (2,3,7) triangle), the reflection domains (images of this domain under the group) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group (reflections in the lines of a given fundamental triangle give a set of 3 generating reflections). This tiling is a quotient of the order-3 bisected heptagonal tiling of the hyperbolic plane (the universal cover of the quartic), and all Hurwitz surfaces are tiled in the same way, as quotients.

This tiling is uniform but not regular (it is by scalene triangles), and often regular tilings are used instead. A quotient of any tiling in the (2,3,7) family can be used (and will have the same automorphism group); of these, the two regular tilings are the tiling by 24 regular hyperbolic heptagons, each of degree 3 (meeting at 56 vertices), and the dual tiling by 56 equilateral triangles, each of degree 7 (meeting at 24 vertices). The order of the automorphism group is related, being the number of polygons times the number of edges in the polygon in both cases.

24 × 7 = 168

56 × 3 = 168

The covering tilings on the hyperbolic plane are the order-3 heptagonal tiling and the order-7 triangular tiling.

The automorphism group can be augmented (by a symmetry which is not realized by a symmetry of the tiling) to yield the Mathieu group M₂₄.

Corresponding to each tiling of the quartic (partition of the quartic variety into subsets) is an abstract polyhedron, which abstracts from the geometry and only reflects the combinatorics of the tiling (this is a general way of obtaining an abstract polytope from a tiling) – the vertices, edges, and faces of the polyhedron are equal as sets to the vertices, edges, and faces of the tiling, with the same incidence relations, and the (combinatorial) automorphism group of the abstract polyhedron equals the (geometric) automorphism group of the quartic. In this way the geometry reduces to combinatorics.