alternative algebra -- featuring the octonions!

Non associative algebra is such an unfamiliar field and it always surprised me how important associativity is to us

I’d love to see a video above Cayley–Dickson construction! As always thanks again!

16:50 what i find really cool about this diagram is that it contains the diagram of quaterions inside it (e1 - e2 - e4) which is it's subalgebra

14:31 lol, I also do that “= … =“ sometimes when I’m taking notes and there’s a long tedious calculation that isn’t very illuminating to my understanding and is just part of the process. I also sometimes use “=> … =>” for the same reasons.

Professor Penn is a teacher in the class pedagogy hard to find elsewhere. His style of delivery hits the heart of motivated learners. Master of ab-initio teaching from basics of Number Theory.Can't go anywhere else to learn about happiness.

Newer developments in "Clifford-Algebra" is a gamechanger for maths. It makes hard problems look simple. They should teach this in school.

Speaking personally a Cayley-Dickinson Construction seems a lot sweeter than gamma reflection via double and contour integration so yes please! More on C-DC

Associativity is one of those properties that you see everywhere but never question why its so important, wonderful video

Definitely want to see the Cayley-Dickinson construction, especially the sedenions, the gradual loss of the usual properties of multiplication, and what, if any, uses there are (in wider mathematics) for the Cayley-Dickinson algebras of still larger dimensions.

I've been fascinated by quaternions for a long time, but octonions have always intimidated me. I appreciate the brief introduction to octonions given here! And I think a video on the Cayley-Dickson construction would be interesting, too!

A series on Geometric (Clifford) Algebra would be great.

Would be nice to see a brief discussion on sedenions and power-associativity.

Great video!! Would love to see connections between octonions and superstrings. And, yes, Cayley-Dickenson as well.

yes please. the stack of increasing dimensional numbers has intrigued me for decades, ever since I learned to use quats for representing movement in R^3. I would also like to see how to instantiate octonions/sedonians, in order to embed quats, complex, &cet inside the higher-order algebras. IS it sufficient, for example to define i ,j, k by e_i ,e_j, and e_k where none of the subscripts are equal?

If you start with Z2 (boolean algebra) and quotient with x(x+1)=1, you get F4. x(x+1)=1 means x and not x is true, in other words F4 is a multivalued logic where the two new elements are x and not x, and their product is true. 4-valued logics are a bit of a jump, because you lose monotonicity, but you can identify x and x+1, they are isomorphic, and the result is the 3-valued logic RM3, after a bit of extra algebra involving the tensor-hom adjunction. It's the same sort of progression as R => C => H => ... except D1, D2, D3, and D4 are the only lattices (D1 is trivial) that form interesting logics. Beyond that you lose more properties and it becomes more complex. But you can still make infinite valued logics. RM3 it should be noted is the famous "yes, no, maybe" logic that is taught in grade school (what, your school didn't teach that? Write them an email).

I'm interested in the Cayley-Dickson construction along with any more Octonian and Sedonian content. This is the first time I've met them.

I’d love to see a video on the Cayley-Dickenson construction!

It would be interesting to compare the Cayley-Dickson construction with the related Clifford algebras.. You can get Complex Numbers and Quaternions either way, but the Octonions aren’t a Clifford algebra because of the non-associativity

With commutative properties, we usually call things anti-commutative if when swapping the two factors you pick up a minus sign, like how in the quaternions you have ij = k but ji = -k. WIth the octonions, it seems like (e[i]e[j])e[k] = -e[i](e[j]e[k]). Are there alternative algebras that aren't "anti-associative" like this?

loved this video :) nice to see stuff outside undergrad