I'm always struck by instances in math unexpected symmetry. For instance, consider ways of coloring the tiles of the hexagonal tessellation of the plane with n colors so that no two tiles of the same color share an edge. One way to do this involves 7 colors. It could be like this: 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 (and repeat). So here's a tessellation by 6-sided figures that is colored with 7 colors, each color forming a subset of the hexagons isometric to that of any other color. Or consider the number 73. Not particularly conducive to symmetry, you might think. But recall the regular 4D polytope called the 24-cell: It's 24 octahedra arranged face-to-face to form the 3-dimensional sphere S^3. In fact it has 24 vertices, 96 edges, 96 2-dimensional faces, and 24 3-dimensional octahedra. This makes it self-dual. So if you draw the vertices and edges of this 24-cell on the 3-sphere S^3, you can also draw the *dual* vertices and edges as well — they will form a disjoint 1-dimensional figure in S^3. Now if you removed both of these "1-skeleta" from S^3, the portion of S^3 that remains is topologically equivalent to the product of a surface M_g of genus g with the real line R. And it turns out that the genus of g M_g is 73. Even nicer, it turns out that the surface of genus 73 that is halfway between the two dual 1-skeleta in S^3 is a *minimal surface*, meaning that if you drew a small simple closed curve C about any of its points, the surface minimizes the area among all surfaces in S^3 whose boundary is the curve C. Even nicer yet, this minimal surface M_73 has an isometry with itself for any isometry of the 24-cell. One of the octahedra can be taken to any of the 24 octahedra, and one of its triangles can go to any of the 8 triangular faces in any of 6 dihedral way, making for 24 x 24 x 2 = 1052 isometries. Nicest of all, the two halves of S^3 created when M_73 is removed can be interchanged, which carries M_73 to itself, giving it another 1052 isometries for a grand total of 2104 altogether. Who said 73 wasn't symmetrical. —Dan