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John Baez (Mar 15 2025 at 01:15):A while back I wrote Some thoughts on the number 6, but James Dolan raises enough points here that I should probably completely revise what I wrote, after I understand this:
pgl(2,f_5) and 5!
i guess that my reaction to hearing about two groups of the same size has varied somewhat over the years; initially tending to be too optimistic that they're isomorphic but then after getting burnt a few too many times maybe the pendulum swang too far in the pessimistic direction .... but anyway, i think i remember now how these two groups are in fact isomorphic not only to each other but also to the coxeter group for the dodecahedron, and i just want to run through the story a bit ....
so, the number six is a bit anomalous in that the group 6! has (for some mystical reason) an extra outer automorphism of outer order 2. that means that in addition to the tautological answer "elements" to the question "what has a 6-element set got 6 of?" there's a more exotic answer. and it turns out that the more exotic answer is "projective line structures". that is, it turns out that the groups 5! and pgl(2,f_5) (that is, the stabilizers of a point and of a projective line structure, respectively) aren't just abstractly isomorphic, they're in fact permuted up to conjugacy as subgroups of 6! by the outer involution.
(sometimes i like to think of a "projective line structure" on a set as a quaternary "cross-ratio" operation valued in some particular standard projective line, in this case "f_5 p^1", to make things a bit more concrete ....)
so then how does the dodecahedron fit in here? going on guesses guided by vague memories of discussions about sylvester's work on "synthemes" and so forth, i'm thinking maybe there's a canonical equivalence relation on the vertexes of a regular dodecahedron given by "belong to the same inscribed tetrahedron", dividing the 20 vertexes into 5 equivalence classes of size 4 each, and that from an arbitrary 5-element set you can functorially reverse-engineer a regular dodecahedron of which it's the set of inscribed tetrahedrons ....
but there's still lots of things that somewhat confuse me here. like, i seem to get an answer to the question "what has a 5-element set s got 6 of?" by taking the exotic elements of the 6-point sets s+1; but is that the same as the opposite-face-pairs of the dodecahedron reverse-engineered from s? my intuition seems to have trouble with this so far ....
actually i seem to be getting that there's a contrast between tetrahedrons inscribed in a dodecahedron in a left-hand vs in a right-handed way .... no idea what that means ....
also i think i've neglected so far to explicitly incorporate into the story the idea that the dodecahedron is the modular curve x(5).