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I get how to construct number systems: $\Bbb{N,~Z,~Q,~R,~C}$. But what about the hypercomplex? How do we go from $\Bbb C$ to $\Bbb H$ and $\Bbb O$?

I've also heard it stated that the octonions are the "center" of mathematics. If that's true, then what would be a good way to explain or depict that graphically?

I took the liberty of adding the crucial part of the interview within the linked video in the question body to make it more easy to understand the part where the OP said that he heard "the octonions are the 'center' of mathematics".

Eric Weinstein:

"If I were running the NSF I would say why are we not putting money into there four very strange objects called $F4$, $E6$, $E7$ and $E8$ to try to figure out what they are telling us. We know, that they are at the center of mathematics" [emphasis by me]

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QWERTY_dw
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    I never heard that Octonions are the "center" of mathematics in any context. However, regarding the construction: see here and then here. – mrtaurho Jul 15 '19 at 19:24
  • @mrtaurho https://youtu.be/Q0QtZmLC14U – QWERTY_dw Jul 15 '19 at 19:33
  • It seems to arise when one constructs exceptional simple Lie groups, such as in https://people.ucsc.edu/~weissman/WhatIsG2.pdf ... But I dare not say more. – Hw Chu Jul 15 '19 at 19:57
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    @user40358 I have watched the whole interview now (thanks for the link!) but either I missed the point where he actually states that 'octonions are the "center" of mathematics' or it was not there. The only connection I can see at all between this interview and octonions is via Lie Groups and Lie Algebras as he is talking about the $E_8$, which has a close connection to Lie Algebras too. As I am far from being an expert in this field I can not evaluate the importance of this - from my point of view loose - relation between them. – mrtaurho Jul 15 '19 at 20:29
  • @mrtaurho He says it at about 2:40. – QWERTY_dw Jul 15 '19 at 20:33
  • @user40358 Well, to be precise he says "we know that they are (the groups he named before) at the center of mathematics". I added the crucial part of the interview to your question. – mrtaurho Jul 15 '19 at 20:47
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    https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction – md2perpe Jul 15 '19 at 21:52

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