the octonions!

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[SimonClarkstone]

"Greetings fellow numbers. I am just a number like you and certainly not two quaternions in a trench-coat. Let us discuss how our multiplication operators are associative. See how mine obeys left-alternativity and right-alternativity; surely a sign of associativity. Wow; look at what time it is already. I must hurry to pick up my… uh… matrix representations from school. Goodbye."
*Departs, followed by the Sedenions, who have to repeatedly stop and pick up the zero divisors that keep falling out of their sleeves.*

[alextaunton]

Pls do a full length video doing some math with octonions, there literally isnt any such video on youtube

[kylecow]

why is litterally everything in maths secretly the fano plane

[djttv]

I would be interested in how it came about. Why was it developed? What motivated its development, etc. Does it have real world applications?

[doctorrodman]

Wow, Jordan’s Algebra truly is exceptional

[NbuGry]

Dude forgot to have arrows pointing from the outer three e's to each other.

[alexanderclose]

My master's dissertation "Octonionic Aspects of Supergravity" can still be found online (search with Alexander Close Imperial College). The octonians can encode 8 dimensional particles and decomposing them into quartonians and complex numbers etc corresponds to dimensional reduction on the torus. Their non-associativity means their automorphism group is the exceptional Lie algebras G_2 !

[Tim-Kaa]

Ah yes I remember good ol days watching Kohl Furey's videos on octonions. Please post more vids, it's a fascinating subject.

[yurenchu]

Alright, I've figured something out to make the diagram more regular: if e₅ and e₆ swap places, and e₃ and e₇ swap places, and we adjust the arrows to match the new positions, then the new pattern of the arrows will show a rotational symmetry (which it doesn't have now).

The yellow arrows will all point counter-clockwise along the sides of the (large) triangle, and the pink arrows will all point along the median lines in the direction from triangle vertex to the midpoint of the opposite side. (The blue arrows remain unchanged in their counter-clockwise direction.)

[Tehom]

The octonions are fascinating. I assume you are familiar with (my friend) John Baez's work on them; if not, definitely give it a look. There's tons to say about the octonions and I hope you will make a longer video, if you want to.

[yurenchu]

Why does this diagram remind me of ? :-)

I'm familiar with quaternions but not with octonions, so this is new for me. The diagram looks kinda confusing though, I'm never going to memorise which indices are on which vertices, and which directions the arrows are pointing (because the diagram doesn't show symmetry; it looks too arbitrary). Maybe there's an alternative diagram that presents their order in a clearer, more symmetrical way, but I haven't taken the time to figure that out yet. So far, it seems to me the seven "loops" are better presented by simply writing them out:

e₁ → e₂ → e₄ → e₁
e₂ → e₃ → e₅ → e₂
e₃ → e₄ → e₆ → e₃
e₄ → e₅ → e₇ → e₄
e₅ → e₆ → e₁ → e₅
e₆ → e₇ → e₂ → e₆
e₇ → e₁ → e₃ → e₇

Each loop contains three elements, each element is member of three loops (which makes sense because in each loop an element "meets" two other elements so it meets all other six elements in three loops), and the indices per loop are (n+1, n+2, n+4) in a "modulo 7" cyclic manner (more precisely: {(n, n+1, n+3) (mod 7) + 1}, for n = 0, 1, 2, 3, 4, 5, 6 ).

Note: the indices per loop are congruent to (k+2⁰, k+2¹, k+2², k+2³, k+2⁴, ... ) modulo 7, for k = 0, 1, 2, 3, 4, 5, 6 .

[fuseteam]

I __really__ can't see the pattern in the elements placements

[sharpnova]

every time you go up a level in the set of vectors you lose a symmetry. quaternions retain associativity and that's what you lose when you go to the octonians

[jasminsainiplan]

In this pyramidic scheme, where’s the PEMDAS law?

[MaxxTosh]

I know you said it’s non associative, but is the choice of products given (up to isomorphism) uniquely the only way to be anti-associative, such that x(yz) = -(xy)z?

[leif]

this video isn't totally clear. if all the elements are the square root of negative qmmthen they are all the SAME..so you really just have one element..so this video doesn't make sense..why say you have seven different elements when you don't..and I forgot what the fano plane is..and it's not clear what he means when he says it spans 1 plus these other elements..is the number 1 an octonion??