Higher Genus “Soccer Balls” Picture Page
Soccer ball pattern construction
A “soccer ball pattern” on a surface is constructed from a kaleidoscopic tiling on a surface as follows. Let S be a surface of constant curvature. The sphere is the only (constant curvature) example in our ordinary 3D experience. For a triple of integers (l,m,n), let T be an (l,m,n)-triangle on S, i.e., triangle with angles of Pi/l, Pi/m, Pi/n. The triangle defines a kaleidoscopic tiling on the surface by repeated reflecting the tile in its edges. This tiling has been denoted the red dashed lines in all of our figures below. The tiling group G*, generated by the reflections in the sides of T, gives us a unique labeling system for the tiles (triangles) of a tiling, i.e., each tile is of the form hT for some unique h in T. Thus, there are |G*| tiles on the surface S. If g is the genus of S then, by an Euler characteristic calculation,
4g-4 = |G*|(1/l+1/m+1/n-1).
Whether the tiling occurs on a sphere, torus, or higher genus surface depends on whether 1/l+1/m+1/n is >1, =1 or <1.
Let P be a point anywhere in T, including on an edge or at a vertex. Now create the set of points hP for h in G*. Connect two points if there are mirror images of each other in the edge of some triangle. In the figures these lines are black. For the interior, the edges and each vertex there is a unique point among all the choices such that the resulting pattern consist only of regular polygons. For the interior of a triangle we pick the incentre of the triangle, it is equidistant to all sides. If l, m and n are distinct there are seven possible distinct patterns with regular polygons. If an interior point is chosen then polygons have 2l, 2m and 2n sides respectively. If P is on a edge or at a vertex some of the polygons change to a regular polygon on half the number of side or disappear entirely. Special rules may apply if one of l, m or n equals 2.
The soccer balls for the sphere are in Table 1. In this case we must have 1/l+1/m+1/n > 1. Note that the soccer ball pattern is indeed realized for the P on the edge of (2,3,5)-triangle. Also every one of the platonic solids – tetrahedron, cube, octahedron, dodecahedron, icosahedron – (or rather their projections to the sphere) is realized by selecting an P on an vertex. Indeed the symmetry descriptions (tetrahedral = (2,3,3), octahedral = (2,3,4), icosahedral = (2,3,5)) are obtained by taking the full symmetry group of the corresponding Platonic solid. Also represented are the truncated regular polyhedra (symmetrically cut off the corners of a regular polyhedron). There is also family of dihedral symmetry types with (2,2,n)-tiles, but we have not reproduced them here.
If 1/l+1/m+1/n = 1 then the corresponding surface must be on a torus. The relevant tilings are (2,3,6), (2,4,4), and (3,3,3). Examples of the corresponding “soccer tori” are given in Tables 2.a, Tables 2.b, and Tables 2.c, respectively. The corresponding surface, when embedded in ordinary space, does not preserve the angle characteristics as happens with the sphere. However the characteristics are preserved if we unwrap the tiling or soccer ball pattern to a pattern on the plane. Indeed there is a map, q: plane –> torus, mapping the tiling and soccer ball patterns to corresponding patterns on the torus. Thus, one may attack the problem by first tiling a portion of the plane –called a fundamental region — and then identifying opposite edges to construct a torus. The lifted patterns on the plane are given along side the patterns on the tori in the tables noted above. These plane soccer tori give us a number of symmetric tilings of the plane by regular polygons. For each possible tiling there is an infinite family of groups G*, and hence tilings,of order kN2 , where N is an integer and k is twice the largest of l, m and n.
Genus >= 2
If the surface has genus 2 or higher then 1/l+1/m+1/n < 1. There are infinitely many solutions for this and so the classification of higher genus soccer patterns is not complete in this case. Indeed here is a table of the number of triangular tilings as a function of the genus. (see the preprint tileclass.pdf on the tilings.org website for classification).
As in the case of a torus, there is no conformal model in three dimensional space that perfectly captures the angle an length relations for the tiling or the soccer pattern. Moreover one cannot represent the symmetries of the soccer pattern by rotations and reflections of the surface as we can with the sphere. Thus one resorts to constructing a tiling or soccer pattern in the “plane” and then taking a finite portion of a tiling or soccer pattern in the plane and identifying “opposite” edges to form a surface. But now, the plane is the hyperbolic plane and the portion of the plane to be selected is determined by the structure of the group G*. The most complex example for genus 2, has a tiling by 96 (2,3,8)-triangles. By picking P in the soccer ball pattern construction to be a Pi/3 vertex of the triangle, a soccer ball pattern consisting of 6 regular, octagons with an angle of 120o ‘s at each vertex. The (2,3,8)-triangles may be joined in pairs along an edge to form a (3,3,4) tiling. This is the same relationship between the (2,3,6) tiling and the (3,3,3) tiling. Pictures of (partial) tilings, fundamental regions and a description of the rotation group for these tilings is given in the following table. The construction of the identification of sides and the soccer patterns is left to the interested reader.
|(l,m,n)||G and |G|||2|G| = # tiles on S||tiling||fundamental region|