Klein Quartic - 3-dimensional Models

3-dimensional Models

The Klein quartic cannot be realized as a 3-dimensional figure, in the sense that no 3-dimensional figure has (rotational) symmetries equal to PSL(2,7), since PSL(2,7) does not embed as a subgroup of SO(3) (or O(3)) – it does not have a (non-trivial) 3-dimensional linear representation.

However, many 3-dimensional models of the Klein quartic have been given, starting in Klein's original paper, which seek to demonstrate features of the quartic and preserve the symmetries topologically, though not all geometrically. The resulting models most often have either tetrahedral (order 12) or octahedral (order 24) symmetries; the remaining order 7 symmetry cannot be as easily visualized, and in fact is the title of Klein's paper.

Most often, the quartic is modeled either by a smooth genus 3 surface with tetrahedral symmetry (replacing the edges of a regular tetrahedron with tubes/handles yields such a shape), which have been dubbed "tetruses", or by polyhedral approximations, which have been dubbed "tetroids"; in both cases this is an embedding of the shape in 3 dimensions. The most notable smooth model (tetrus) is the sculpture The Eightfold Way by Helaman Ferguson at the Mathematical Sciences Research Institute in Berkeley, California, made of marble and serpentine, and unveiled on November 14, 1993. The title refers to the fact that starting at any vertex of the triangulated surface and moving along any edge, if you alternately turn left and right when reaching a vertex, you always return to the original point after eight edges. The acquisition of the sculpture led in due course to the publication of a book of papers (Levy 1999), detailing properties of the quartic and containing the first English translation of Klein's paper. Polyhedral models with tetrahedral symmetry most often have convex hull a truncated tetrahedron – see (Schulte & Wills 1985) and (Scholl, Schürmann & Wills 2002) for examples and illustrations. Some of these models consist of 20 triangles or 56 triangles (abstractly, the regular skew polyhedron {3,7|,4}, with 56 faces, 84 edges, and 24 vertices), which cannot be realized as equilateral, with twists in the arms of the tetrahedron; while others have 24 heptagons – these heptagons can be taken to be planar, though non-convex, and the models are more complex than the triangular ones because the complexity is reflected in the shapes of the (non-flexible) heptagonal faces, rather than in the (flexible) vertices.

Alternatively, the quartic can be modeled by a polyhedron with octahedral symmetry: Klein modeled the quartic by a shape with octahedral symmetries and with points at infinity (an "open polyhedron"), namely three hyperboloids meeting on orthogonal axes, while it can also be modeled as a closed polyhedron which must be immersed (have self-intersections), not embedded. Such polyhedra may have various convex hulls, including the truncated cube, the snub cube, or the rhombicuboctahedron, as in the small cubicuboctahedron at right. The small cubicuboctahedron immersion is obtained by joining some of the triangles (2 triangles form a square, 6 form an octagon), which can be visualized by coloring the triangles (the corresponding tiling is topologically but not geometrically the t_0,1{4, 3, 3} tiling). This immersion can also be used to geometrically construct the Mathieu group M₂₄ by adding to PSL(2,7) the permutation which interchanges opposite points of the bisecting lines of the squares and octagons.