Here we see a regular dodecahedron. (These don't look great as static images, I know, but the program allows you to rotate the figures which makes them a lot easier to discern.) Suppose we were to twist two of the opposite pentagonal faces; then we'd get something that looks a little like this:
In case you couldn't tell from the shape, the background has turned light yellow to indicate that this permutation is not in fact an automorphism of the dodecahedron. This is a platform independent Java application, and I'll post a download as soon as my web hosting comes back up. If anyone would like to look over my Java code and tidy it up, and/or make it into a workable applet, that'd be great; I really don't know Java and this took me way longer to write than it should have.Relevant math question: If we let S6 act on the vertices of the icosahedron, how many different graphs (up to isomorphism) do we get?
Update: now available for download. Extract the archive into a directory, then either double-click on the .jar file or use the provided shell script or batch file depending on platform. The graphs/ directory contains some different graphs you can load; if you peek at one of the files in a text editor it should be pretty clear how to make your own. Enter permutations in cycle notation, e.g. (1 2)(3 4), and beware the following caveats:
- The vertices are numbered 0,1,...n-1 rather than 1,2,...n
- You don't have to reduce your permutations, so you can enter e.g. (1 2)(1 3)(3 4 5) or somesuch. Be aware, though, that the program composites permutations backwards, so you'll have to flip things right to left with respect to normal mathematical notation.
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