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I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a little less vague, I should indicate what I mean by "real world": First of all, in physics, I require that this be experimentally verifiable, so $G_2$ holonomies in stringy math should not count, nor should grand unified $E8$-theories. Industrial applications would be great, as would examples from biology.

gmvh
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Lorenzo Del Vecchiopontopolos
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    Do uses of the E8 lattice in error-resistant communications count? – Noam D. Elkies Sep 09 '23 at 11:34
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    That sounds very interesting! So yes! – Lorenzo Del Vecchiopontopolos Sep 09 '23 at 11:55
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    The alcoves of the affine $G_2$ root system give a highly symmetric tiling of the plane by congruent triangles. I would be amazed if this picture didn't appear somewhere "in the real world." – Sam Hopkins Sep 09 '23 at 12:58
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    Baez and Huerta - $G_2$ and the rolling ball. – LSpice Sep 09 '23 at 13:33
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    I have no idea what you mean by the “real world”, but whatever it is math is part of it. – Andy Putman Sep 09 '23 at 14:38
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    @AndyPutman: I tried to define "real world" in the question. – Lorenzo Del Vecchiopontopolos Sep 10 '23 at 10:29
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    @AndyPutman How many times have I heard a mathematician say "I have no idea what you mean by X," or slight variations of it, where X is just a perfectly plain concept understandable by everyone. Typically they are pretty smug about it, too. But if you really want a fruitful conversation with someone, you should at least attempt to construct a generous interpretation of what the other person says. – R.P. Sep 17 '23 at 16:35
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    @R.P.: In light of the thousands of years of philosophical reflection on the subject, I don't think the nature of the "real world" as it relates to mathematics is a "perfectly plain concept understandable by everyone". Being some kind of Platonist, I think that mathematical concepts are perfectly real and exist in the world. When my students ask me a question like that of the OP, I always interpret it as "When will me or someone like me need to know this?" and I answer accordingly, but on a website for professional mathematicians I think we can expect more rigor and precision in our language. – Andy Putman Sep 17 '23 at 17:20
  • @AndyPutman Okay. Thank you for being kind enough to clarify your response. :) – R.P. Sep 17 '23 at 17:41
  • $E_8$ appearing in an experiment: https://mathoverflow.net/questions/32315/has-the-lie-group-e8-really-been-detected-experimentally – Skip Jan 15 '24 at 13:42

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As Noam Elkies alluded to in a comment, the E8 lattice plays a role in coding theory, basically because it such an efficient sphere packing. For example, Kurkoski has proposed using it for error correction in flash memory: The E8 Lattice and Error Correction in Multi-Level Flash Memory (but I don't know if this error-correction scheme is actually used in practice).

Timothy Chow
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