Computing Symmetries

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‘‘One of my interests is the study of regular maps, which began with the platonic solids - regular convex polyhedral known to the ancient Greeks. If you
expand each solid so that it becomes round, then its edges and vertices can be thought of as a highly symmetric network drawn on a sphere. Regular maps generalise platonic
solids to surfaces of higher genus, like a torus or double torus. “The more handles you stitch on, the higher the number of smooth holes that appear, which gi
ves the
genus - the genus of a sphere is 0. A PhD student and I used computers to determine all regular maps of genus 2 to 15, then five years ago with new computer methods I was able to extend it to genus 100, and in July I got up to genus 300.”

Computers provide huge amounts of data, says Conder. “The data make it possible to see patterns that go on to infinity, and led me to the answers to some long-standing questions about regular maps.” One was about chirality - when the map is different from its mirror images. On surfaces of genus 0 and 2, 3, 4, 5, and 6, there are no chiral regular maps. “I was able to prove this happens for infinitely many genera.”

Other situations involving maximum symmetry that interest Conder include graphs, Riemann surfaces, hyperbolic
manifolds, and closed non-orientable surfaces, in which inside and outside (or left and right) cannot be defined. The most well-known such surface is the Klein bottle.

“A question about structures on such surfaces is this: if you know it has a certain degree of symmetry, does it have more? Some colleagues in Spain and I solved this problem a few years ago for orientable surfaces - the sphere, torus, double torus and so on. Now I’m interested in doing the same thing for non-orientable surfaces.” Sometimes pure mathematicians make accidental discoveries, such as one which Conder happily stumbled upon five years
ago. “You take a network, and prescribe in advance the maximum degree - the number of nodes any one node can be joined to, and the diameter - the largest number of steps needed to get from one node to another.” “For example, a cube has eight nodes, with three steps from one corner to its opposite, so diameter 3.” Often the most efficient
networks, which are interesting to computer scientists and engineers, have the largest amount of symmetry.

Conder used computers to find all the symmetric networks of degree 3 with up to 2,000 nodes. When he checked their diameter, he found a new one that is now the largest known network of degree 3 and diameter 10. He has since extended the census up to 10,000 nodes.

His most recent work is on polytopes, which are like regular maps in other dimensions. The smallest regular 2-D polytope is an equilateral triangle, with type {3}, and the smallest regular 3-D polytope is the tetrahedron, with type {3,3}: every node is on three edges, and every face has three edges. “A few people expected in that in higher dimensions the type would consist of all 3s. But the surprising thing is that from rank 9 onwards, the type of smallest example has all 4s.”

“Some mathematicians don’t like using computers to solve problems. But I find them really helpful in producing data, which often reveal patterns. Then I can try to prove that the patterns exist universally - beyond the data. This approach has shown me the way to a lot of general discoveries that have been surprising or unexpected, and I doubt would have been possible otherwise.”


Top: A {3,7}-tessellation of the hyperbolic plane.
Middle: Hands are chiral - the mirror image cannot be superimposed on the original.
Bottom: A Klein bottle, a closed nonorientable surface.