The Octonion Math That Could Underpin Physics

I would say that studying octonions exclusively would be something I personally would avoid, as I would rather try to study the four structures (reals to octnonions) together, either more generally (e.g., group theory and ring theory) or more abstractly (e.g., as members of categories) and form an opinion on whether I think octonions in particular are useful for the questions that I want to ask.

That is not to say that the research here is not interesting, but it is difficult to judge that from "popular" mathematics articles. I got the impression that the author of the article places a much higher priority on the pictures accompanying the post.

I remember that I found it interesting that studying the four dimensional spacetime bears more fruit than stopping at three space dimensions, and that at the same time from complex numbers the next structure ends up also having four dimensions (i.e., being modelled by 4-tuples). However, apart from being interesting in this narrow sense, I do not know whether this suggests any creative yet precise mathematical questions.

So this is kinda roundabout however there’s homotopy type theory which states that topology=logic=type theory. For a second let's say that theres a fourth "element" in this equation and that's quaternions (this isn't a stretch I can provide some papers that hint at this).

Linear logic is classical logic with the added reference to time without explicitly referencing time.

Dual quaternions all about modeling deformation over time.

Therefore, I would conclude that linear logic and dual quaternions are related.

An "algebra" is a set of thingies that have binary operations defined on them (i.e., operations that take two thingies as input and output another thingie). The thingies are usually referred to as "elements" of the algebra. Any algebra will at least have the "add" and "multiply" operations, but might also have others.

A "division algebra" is an algebra where the operations are add, subtract, multiply, and divide, and all of them work basically the same as they do with ordinary numbers (but they won't have all of the same properties--see below).

A "normed division algebra" is a division algebra with one additional operation called "norm", that takes as input any element of the algebra and outputs a real number, the "norm" of that element. On the ordinary real numbers, the norm is just the absolute value.

The "over the real numbers" part means that you can construct the normed division algebra by starting with real numbers; or, to put it another way, all of the elements of the algebra are "made of" real numbers. The simplest way of viewing this is as a repeated operation of pairing: complex numbers are made of pairs of real numbers, quaternions are made of pairs of complex numbers (hence sets of four real numbers, hence "quater"), and octonions are made of pairs of quaternions (hence sets of eight real numbers, hence "octo"). But each step in this series loses a key property. The reals are totally ordered; the complexes are not. The complexes are commutative under multiplication; the quaternions are not. The quaternions are associative under multiplication; the octonions are not.

> Is there any place that explains mathematical concepts in ... different ways?

Not really, because the only way people have found to really understand mathematical concepts is to build them up out of simpler mathematical concepts. That means you can't just encounter a complicated mathematical concept and expect to understand it if you don't understand all the simpler concepts it is built from. There are no shortcuts.