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See: Math, Tetrahedron

Hi Kirby,

I was reading about Bott-periodicity and found another of John Baez's posts. He writes about triality: http://math.ucr.edu/home/baez/week61.html

The Wikipedia article is helpful: https://en.wikipedia.org/wiki/Triality

The Dynkin diagrams describe how to relate the reflections that generate a group. These reflections are highly relevant in thinking about the Platonic solids, for example.

These relations between the reflections are highly constrained, especially as the dimension of the space grows large. Basically they are chained together in a long line with at most one short "leg" sticking out at or near the beginning. But in the particular Dynkin diagram for D4 the line is so short that it equals that leg as in the picture. (F4 is another exception). See: https://en.wikipedia.org/wiki/Dynkin_diagram

So the particular D4 looks like your coordinate system for the tetrahedron.

I didn't understand everything but I enjoy and appreciate John Baez's posts because he gives an inkling that there is a big picture in math, or at least, pieces of it. That kind of immersion leads in time to fluency, I think. It's a bit opposite to the standard "step-by-step" approach.

One of the reasons that Lie groups and Lie algebras are important is because they link together the "calculus world" (Lie groups are "differentiable manifolds") and the "discrete world" (Lie algebras are based on "root systems" that are geometric reflections). The calculus world is the "exponential" of the discrete world, at least I've seen formulas like that.

Also, John wrote about the Clifford algebra periodicity here which is somehow related to Bott periodicity: http://math.ucr.edu/home/baez/week82.html

Here's a post where John Baez directly links "triality" and the exceptional Lie group E8: http://math.ucr.edu/home/baez/week90.html E8 is the one that surfer-physicist Garrett Lisi is famously working on. Here' his TED lecture on that: https://www.ted.com/talks/garrett_lisi_on_his_theory_of_everything?language=en

#### Triality

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