I really need the link. I want to see its application...

Need the link

This was it. Not sure what's harder to read, the math or the broken English! arxiv.org/PS_cache/arxiv/pdf/0704/0704.0136v2.pdf

Geometric Algebra for n-dimensional space has 2^n coefficients in its objects; so it gets quoted as 2^n "dimensions". 5D space for conformal algebra is common, which is 2^5 dimensional. The 2^n comes from n directions in space included or not; because directions in space square to real numbers -1, 0, or 1.

Crazy. Wonder how long time it would take to understand än article like that.

Same 😂

Check out the channel dialect. You will get an intro to vector calculus. They also demonstrate elementary matrix algebra. They just started a series on Christoeffel tensors so if you can get Riemannian geometry, you're well on your way to getting quaternions, it's just adding more to matrix operations.

@@philipm3173 is there a channel to explain anything of what you just said?

@@sazam974 3Blue1Brown

@@sazam974 they have a 16 part course called the essence of linear algebra which introduces vectors and linear transforms.

howw

I did the same in 9th grade

Like come up with the multiplication rules? Or something else?

@@schizoframia4874probably the multiplication rules

Was there a bridge nearby ?

Zorn matrices might be nice. Yes, that Zorn.

So splithy

split hyper complex?

@@pugza1s731 yes

Yes, especially a way to systematically construct splithypercomplex numbers over any field, including those of characteristic 2. This most likely means not all independent generators can be anticommuting but instead may be "skew commuting" rather.

The first telephone company in Hamilton Ontario was started by a physicist. He built a young ladies dormatory on the 2nd floor of the exchange because he knew Hamiltonian operators do not commute.

After the sedenions, your balls fall off

Lmao

I kinda hope it all unravels into complete anarchy as you move up through the dimensions, and then suddenly assumes strict rules again. Repeat ad infinitum. That would be awesome :)

Yes, this is a very deep question. Is there an infinite family of (increasingly abstruse) algebraic properties, which are incrementally lost as the ladder is climbed? Or, do the (somehow meaningful) algebraic properties completely run out at some point, and as abstract algebras, the higher-dimensional conjugation algebras are all the same? (But then, they are continuous algebras parameterized by 2^n copies of the reals... which in itself is an algebraic property. They are not isomorphic...)

If you're hoping for things to come back being like the wonderful unity of the complex algebra, sorry, that won't happen. Any algebras that does just what the complex numbers do, is itself the complex numbers. Etc. Fortunately, different algebras are different, and life is richer! These things came to life as abstractions, but people have applied them to real-life problems. For example, multiplication by quaternions preserves geometry in 4 dimensions, and thereby, motion and scaling of solid 3-dimensional objects in 1 time dimension. Their non-commutativity reflects the non-commutativity of 3-D rotations.

So long as |a| is quadratic in the components of a. Otherwise matrices provide a counter example.

doesnt this have to do with the topological characteristics of spaces? I've been getting into topology and there are some theorems dealing with parity of dimensions and how they don't allow for certain constructions

On sedenions there are Zero divisors

@@benjaminojeda8094 Yes, and they are not a composition algebra, that is they fail the important rule: forall X and Y: |X Y| = |X||Y|. Where || is the quadratic norm form in question (related to a symmetric bilinear form sometimes called an orthogonal form).

@@Utesfan100 I am pretty sure that, as long as we investigate *composition algebras* rather than merely the special case of *normed division algebras*, there are 3^n dimensional algebra analogues with cubic norms, to these 2^n dimensional algebras with quadratic norms. In the general case these norms are not necessarily positive definite but merely nondegenerate indefinite.

Google "cohl furey"

Such as those used in matter theories.

😢😢😢😢😢

They don't have applications.

There are attempts at applying Octonions to physics. Probably not the beat idea, but there are some interesting results there.

Ummm...I think Newton's insight about gravity and Einstein's insights about relativity (among others) are *just a tiny bit* more famous than that. ;-)

​@@w.randyhoffman1204we are talking about history of mathematics here :)

@@mathophile716Newton’s insight into Calculus then.

If I see you spell it like that, it's on sight.

​@@praharmitra Let's not debate what's greater, Newton, Hamilton, doesn't matter. All true geniuses.

I would like to see this too

The basic idea is that a^-1 * (a*b) ≠ b ≠ (b*a) * a^-1 generally. That is multiplication by the reciprocal of a is not the same as division by a, which means these reciprocals are not true inverses. You need to pick four fully independent sedenions a, b, c and d (none of them expressible through multiplying and/or adding the others). If you pick them to be orthogonal i think you may form (a+b) * (c+d) or something and show that is then zero. I don't remember exactly though, it was some years since i read about it. Iirc there is some claim about the zero divisors of the sedenions being closely connected to the Lie group G_2.

Matrix algebras ate associative which isn't the case for octonions.

Nope, it isn't! There's another process that generates the same 3 algebras at the start, but never loses its associativity, and that is Clifford algebras. The Reals are Cl(0,0); Complex numbers Cl(0,1); Quaternions Cl(0,2). Past that it diverges from the Cayley Dickson construction with Cl(0,3), Cl(0,4), and so on. Like the Cayley Dickson construction, each is 2 times larger than the prior, and in general Cl(0,n) is 2^n dimensional (or rather Cl(p,q,r) is 2^(p+q+r) dimensional). Most explanations of Clifford algebras won't actually define the algebras in this way though, instead generating them from Cl(2), Cl(3), and so on, and then using the even subalgebras to extract the Complex numbers and Quaternions respectively. This method more cleanly shows that the systems in question are specifically the algebras of rotations in 2 and 3D respectively.

Some while ago i tried working out something similar to Clifford algebra, but using the Moufang identities and scalar squaring, anti-commutation and anti-association relations rather than associativity and just scalar squaring and anti-commutation relations. This would produce O from H, but then produce something entirely else from O. Never got the calmness of mind to complete my reasoning though, life is a bitch sometimes.

none at all

I think the second identity you mentioned is this one: en.wikipedia.org/wiki/Euler%27s_four-square_identity

@@jakobthomsen1595 Thanks for the article. It looks as if it's only possible up to 8 variables if you want linear expressions in the squares, but there are analogues for any power of 2 if you allow rational functions.

Thanx to octonions, there is such a formula in 8 dimensions, yet there is none in 16 dimensions or higher.

Mathematics doesn't have to feel elegant to be useful though ;) Of course, we could all just agree that these definitions don't make sense. Though we then might lose some useful applications... I think, grahams number could be seen as not really elegant, but that doesn't really matter, right?

tell that to roboticists extensively using quaternions

​@@Nettlebed7 or game developing where it feels like half of the times I look up something I need to understand quaternions to understand how it works

All geometric algebras are associative, the octionions aren’t! That means the octionions and beyond are not the even subalgebras of some geometric algebra. Depending on your perspective this is either disappointing (a nice pattern doesn’t hold) or exciting (the Cayley-Dickinson Construction is a different way to generate algebras that happens to line up with the Geometric-Algebra-derived ones in low dimensions!)

@@kikivoorburg Thanks for your remark! The next question is where does the equivalence breaks between the 2 algebras ( octonions and the even subalgebra of a 4D geometric algebra with unit metric) . I guess those 2 algebras still share a lot.

What sort of idiot gets the diagram that charts the system wrong?

Well octonions and above are non-associative, but Clifford algebras are _always_ associative, so the isomorphism stops at the quaternions.

Maybe it is more of a coincidence that the more "primitive" structures like complex numbers and quaternions show up in unrelated contructions. And yes, Octonians and above don't fit in any geometric algebra, because of associativity, as was said above. What would the cartesian product be? Addition of elements with different grades? The isomorphic embeddings of R, C and H may just not be related to each other in a way that resembles the Cayley-Dickson construction, because it uses tools not available within one Clifford algebra. Or maybe you would need a very large one to have blades of grades that don't interfere with each other that they become "free" = independent like the components of a Cartesian product, but then you don't actually benefit from Cliffordness.

Imagine one.

Such numbers are called split complex numbers, or rather they are part of those numbers.

Yeah, and there could be a virtual unit, v, so that abs(a*v)=-a. |iv|=-i also, there could be |(-a)v|=a Interesting, isn't it? The general form could be: a+bv Higher orders could entail: a_0+a1v1+a2v2+...+a2^n-1v^2^n-1 Just my speculation.

@@Gordy-io8sb Try conic complex numbers, also called tessarines and bicomplex numbers. They are a commutative algebra over the complex numbers, indeed they are a composition algebra with complex quadratic norm. All complex numbers are represented as norms (or absolute values if you like) of numbers in this algebra.

Sedenants are 4D graphical regions.

The answer I find the most satisfying is from geometric algebra (though it’s far from rigorous): Basically, any time you introduce an element in a geometric algebra, you end up ‘generating’ other elements from the multiplication. Let’s try to construct an algebra with just a single “-1” squaring unit we’ll call “i”. Then we have: i^2 = -1 Which ‘generates’ the scalar values. So actually we can’t have a geometric algebra with only “i”, we need {1, i}. Let’s try another -1 squaring element “j”, giving {1, i, j}. Watch what happens: 1^2 = 1 i^2 = -1 j^2 = -1 ij = ? What does “ij” equal? Well, in geometric algebra, different basis elements anti-commute. Hence: ij = -ji This lets us figure out what it squares to: (ij)^2 = ij ij = -i jj i = ii = -1 So by introducing “j” we also ‘generate’ another -1 squaring element which we can label k. So we have {1, i, j, k} where ij = - ji = k and by extension ijk = (ij)^2 = -1. This is the definition of the Quaternions! Notice that we were able to have the 2D complex numbers and the 4D Quaternions, but not 3D-complex like numbers since it sort of “wants” to generate another element. Sadly the explanation doesn’t extend to Octonions since they’re non-associative and all geometric algebras are associative, but I find it a nice way to understand the case for Quaternions more intuitively at least

I was wondering the same thing, although I hadn't considered the fractal nature. Perhaps they would be half-associative 😂😂 We should invent a system.

At this point I think we might want to ask what limits there are, if any, to the amount of different types of numbers and the possible properties of said numbers that can be logically constructed within the rules of mathematics. To what extent can the set of ALL numbers be comprehended at all?

That is not how it works. There is a correspondency between rotations of vectors, and of spinors though, but you need to study something called Bott periodicity and Clifford periodicity to understand it properly. For real vectors of dimension n, the corresponding spinors can be real, complex or quaternion, and they can be either single spinor or two half spinors. Each complex spinor has a conjugate spinor, these function somewhat similar to half spinors. SO R¹ ≈ Spin R¹ SO R² ≈ Spin C¹ SO R³ ≈ Spin H¹ SO R⁴ ≈ Spin H¹± SO R⁵ ≈ Spin H² SO R⁶ ≈ Spin C⁴ SO R⁷ ≈ Spin R⁸ SO R⁸ ≈ Spin R⁸± SO R⁹ ≈ Spin R¹⁶ SO R¹⁰ ≈ Spin C¹⁶ SO R¹¹ ≈ Spin H¹⁶ SO R¹² ≈ Spin H¹⁶± SO R¹³ ≈ Spin H³² SO R¹⁴ ≈ Spin C⁶⁴ SO R¹⁵ ≈ Spin R¹²⁸ SO R¹⁶ ≈ Spin R¹²⁸±

So for example quaternions are related to 3D and 4D rotations. While octonions (with various internal structures) are related to 5D, 6D, 7D and 8D rotations. 9D rotations are special since they introduce tensoring of previous spinors with 16D spinors, which continue by periodicity onwards.

Yes. There is a relationship between these algebras and projective spacea over the field GF(2) = Z/2Z. This is because every generator unit squares to a scalar.

anything of interest is lost

Last remaining shreds of sanity

That from these honored dead, they take increased devotion to the task for which they gave the last full measure of devotion. 💀

For associativity and commutativity at least, I think the assumption of their holding in the reals is reasonable since that proof is probably a bit out of scope of the video. He proved commutativity in the complex numbers, and that's the last algebra where it holds. He did skip associativity in the complex, but I think the proof of that would look the same as the proof of it in the quaternions, which he did show

I'd recommend Chapter 33 in "The book of involutions", a book by Alexander Merkurjev, Jean-Pierre Tignol, and Max-Albert Knus. But I'm a graduate student mastering algebra, so this might not suit your preferences. In that case I'd recommend "On Quaternions and Octonions". A book by Derek A. Smith and John Horton Conway.

Unit quaternions (isomorphic to SU(2)) generate rotations since they're a double cover of SO(3). Their relationship with the groups SO(3) and SU(2) are among the reasons of many other applications of quaternions to physics. In a similar way, octonions are deeply related to physics and the standard model; for more info I recommend taking a look at: kzfaq.info/get/bejne/pM1hlbCkuNXPl6s.html

Quaternions are used for both 3d and 4d rotations, while octonions may be used (in less obvious ways) for 5d, 6d, 7d and 8d rotations.

Because no higher C-D algebras than quaternions and octonions are composition algebras. Cross product works trivially in one and zero dimensions also.

especially consiering he calls them "sed-OH-nee-ans" instead of "sed-EN-ee-ons"

@@cd-zw2tt What do you call people from Sedona, Arizona?

@@TimothyReeves "John." But he's the only one I know in Sedona. 🙂

​@@cd-zw2ttHmm? Who said onions?

Geometric algebra explains all of this. You can define one by saying how many and what type of basis vectors you want to have. In general G(x, y, z) is saying there are x vectors that square to 1, y vectors that square to -1, and z vectors squaring to 0. G(2, 0, 0) is basically 2d space and isomorphic to the complex numbers. To get something for 3 dimensions simply: G(3, 0, 0) -> 3D space. The rules for working with geometric algebra are very simple and gives as a greater understanding of the objects we use. Find out more: kzfaq.info/get/bejne/bJaqksuhqqfYdGw.htmlsi=87Zw1fA8KWX3Bedu G(4, 0, 0) -> Quaternions G(8, 0, 0) -> Octonions G(2^n, 0, 0) -> 2^n-"nions"

It’s worth noting you _can_ make 3D number systems, they just don’t act like complex numbers. If you demand that the elements of your algebra i and j square to -1 like with the complex numbers, Quaternions, etc. It is (as far as I’m aware) provably impossible to get 3D numbers to work. This, however, does work: 1^2 = 1 i^2 = j j^2 = i i^3 = j^3 = -1 Giving the set {1, i, j} which is 3D and closed under multiplication! It does have zero-divisors though. Look up the video “Let’s invent the Triplex numbers”, that’s where I got this example from. It’s well worth a watch! I think you could define any dimensionality just by having a unit q where q^n = +- 1 This algebra is closed under multiplication and has n dimensions: {1, q, q^2, q^3, … , q^(n-1)} Edit: also, multiplying by i or j in the Triplex numbers both corresponds to some rotation about the diagonal axis (passing through 0 and 1+i+j) so I’m not sure you could do other 3D rotations with it.

@@evandrofilipe1526 the relationship for the Octonions doesn’t actually hold, because all geometric algebras are association and the Octonions aren’t! The Cayley-Dickinson Construction splits off from GA after the Quaternions

There is also a real-imaginary-virtual number system that is 3D, with virtual numbers that have negative absolute values.

@@AlbertTheGamer-gk7sn I always viewed negative numbers as a kind of hack. Like "let's just add one bit of data to numbers and see what happens". But it seems they are quite intrinsic to all of math. I guess i've changed my mind on how i feel about negatives now.

It's arbitrary because it's a definition that was developed historically and then used with that definition after that. I see this type of misconception on KZfaq Math videos, it's just a convention.

@@holliswilliams7717 So, like the tragedy of electrons being negative.

Not really. Higher dimensional rotations are generally much better described using Clifford algebras. In general, the algebra of rotations in n dimensions is Cl⁺(n), or the even subalgebra of the Clifford algebra over R^n. While this aligns with the Complex numbers and quaternions for Cl⁺(2) (aka Cl(0,1)) and Cl⁺(3) (aka Cl(0,2)), they diverge beyond that as Cl⁺(4) only has 6 elements that square to -1 and 2 elements that square to +1, along with being associative but, I think, having zero divisors. If you've read about 4D rotations, it shouldn't actually be that hard to see why things diverge since it's the first time you encounter "double rotations" or rotations that can only be described as 2 rotations around separate axes of rotation, rather than just a single rotation around only 1 rotation axis. This is also why I omitted Cl⁺(1) = Cl(0), since in general, an additional acid of rotation gets added every 2 dimensions. 0 and 1D have 0; 2 and 3D have 1, 4 and 5D have 2, etc. The extra element that squares to +1 instead of -1 is exactly used to represent these double rotations, whereas there are only 6 distinct primitive rotation axes. (1 not-rotation, 6 single rotations, and 1 double rotation.)

@@angeldude101 Also, a torus is a 4D shape that has 2 radii, a doughnut is a 3D cross-section of a torus, that's why it has no beginning, middle, or end.

@@AlbertTheGamer-gk7sn Uh... What‽ A torus is a 2D manifold in 3D space. There's absolutely nothing 4D about it unless you go out of your way to rotate it into the fourth dimension. Spinning a torus around its two separate axis (the linear one through its hole and the circular one through its interior) can act like a 4D double rotation in some ways, but it's just as easily described as resembling 3D screw motion, or, well... Rotation around a circular axis (which is exactly what we initially defined it as).