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Fascinating!
This is an application of group theory that chemists like me are rarely exposed to. We can take a molecule and its 3D geometry and work out which point group symmetry it possesses, or its chemical graph and the corresponding permutation-inversion group.
This post give me some ideas for other ways to explain symmetry in chemistry.
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Very useful for error correction code as well.
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Hi Paul,
thank you very much for this nice notebook. This really makes the article in the quanta magazine twice as valuable.
I only want to add one more sentence, which makes the difference between the two polynomials more transparent, hopefully.
For your function f(x) ( in the article it is denoted with g(x)) the expression of the roots
(a-b)(b-c)(c-a)=\pm 7 is rational and therefore breaks half of the symmetries.
For the other function g(x), the expression of the roots
(r-s)(s-t)(t-s)\notin {\mathbb Q} is irrational and therefore it cannot be used to reduce the number of symmetries. g(x) ends up with the full symmetric group as Galois group.
Finally, when you look at the radical expressions for the roots you can convince yourself, that the solutions for f(x) are simpler than for g(x). While the solutions for g(x) contain nested roots the solutions for f(x) don’t.
Thanks Johannes. I’ve updated the post accordingly—and also linked this Notebook to a recent LinkedIn post by Quanta Magazine.
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