There are many "regular maps on surfaces" — an old-fashioned term for a tiling of a surface by p-sided polygons, q per vertex, with a regularity condition: Any regular map (or chiral map as below) can be made of regular polygons as long as the metric on the surface is adjusted accordingly. * If any polygon is mapped onto any polygon in any of the 2p dihedral ways, that map extends to an isomorphism of the whole structure. If this seems unnecessary, consider a tiling of a torus by just 2 squares next to each other — a 2 x 1 rectangle with opposite sides identified: _________________ | | | | | | | | | | | | ————————————————— (There does, however, exist a regular map of the torus made of just 2 squares.) There is also a weaker condition: * If any polygon is mapped onto any polygon in any of the p rotational ways, that map extends to an isomorphism of the whole structure. If a map on a surface is not a regular but satisfies this condition, it's called a "chiral map". The first and extraordinarily lovely examples of chiral maps occur on tori: There is one made of 5 squares and one made of 7 hexagons. The one of 7 hexagons realizes the 7-coloring of a torus. Its automorphism group is size = 42. A theorem of Hurwitz puts a bound on the size of the symmetry group G of a regular or chiral map. For an orientable surface of genus g: |G| <= 168(g-1). For a chiral map, |G| <= 84(g-1). My favorite regular map is the first one where this bound is reached. It consists of 24 heptagons, 3 per vertex, tiling the 3-hole torus. Its symmetry group is size 336 and its subgroup of orientation-preserving symmetries is the (unique) simple group of size 168, the second-smallest nonabelian simple group. (The smallest one is the alternating group A_5, the group of rotations of the dodecahedron — which is of course an example of a regular map.) I was very surprised when first hearing of this regular map: The 3-holed torus seems to have nothing to do with the number 7, and yet this regular map has a great deal to do with 7. In case you know about complex projective varieties, a 3-holed torus can be expressed as a Riemann surface with exactly this symmetry by the equation X Y^3 + Y Z^3 + Z X^3 = 0 for [X : Y : Z] a point of the complex projective plane CP^2. This is the famous "Klein quartic", about which a whole book was written: <http://library.msri.org/books/Book35/contents.html>. —Dan