John Carlos Baez on Nostr: There was a man who liked small ornamental boxes. He didn't do much with them: he ...

There was a man who liked small ornamental boxes. He didn't do much with them: he just collected them and put them on a shelf. But sometimes he would take one down, turn it over in his hands, and admire it. And one day he noticed one of his boxes had a false bottom that could slide open, revealing... something much more interesting!

I feel a bit like that now. This is the McGee graph. I've known about it for almost 10 years. I've thought a lot about its symmetries. But recently I discovered a surprising fact about the symmetries *of* the symmetries of the McGee graph:

https://golem.ph.utexas.edu/category/2025/03/the_mcgee_group.html

The symmetries of any object form a group. But groups themselves have symmetries, called 'automorphisms'. They come in two kinds. The obvious ones , called 'inner automorphisms', come from the group elements themselves. But there may also be nonobvious ones, called 'outer automorphisms'.

The symmetries of the McGee graph form the smallest group with an outer automorphism that does a good job of pretending to be an inner automorphism, in the following way: acting on any *one* element of the group, this outer automorphism does the same thing as some inner automorphism!

The animation, made by nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqt76as6gjr7pzg0taz40e55smjjegmj89ud7g056aqed90hs7cynsacyu7x (nprofile…yu7x), shows one of the 32 symmetries of the McGee graph.