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The Octonion Math That Could Underpin Physics

quantamagazine.org

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Re: The Octonion Math That Could Underpin Physics

#121

Post by throwaway37585 »

ssivark wrote: post #70
chunky1994 wrote: post #64

It's rather amusing that the author assumes that non-associtiave objects are "weird" for physicists (or at least that was my reading), since the velocity addition formula is in general non-associative and that has been extensively studied. (I remember three separate occasions in my undergrad particle physics class where we actually went through all the calculations involved with the velocity addition formula and fina…

To be fair, one never talks of relativistic "velocities" beyond undergrad physics. It is much easier to talk about Lorentz "boosts", and as members of a Lie Group, they are associative, though not commutative.

This. It's better to just make the full leap into spacetime geometry (4-vectors and 4-tensors) than keep trying to slice it into annoying 3-vectors.

Re: The Octonion Math That Could Underpin Physics

#122

Post by uryga »

PurpleBoxDragon wrote: post #118
sdenton4 wrote: post #13

I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "There are exactly four normed division algebras: the real numbers ($\R$), complex numbers ($\C$), quaternions ($\H$), and octonions ($\O$). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still…

Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a com…

Quaternions, in three words: "Rotations in 3D". I like this explanation: https://probablydance.com/2017/08/05/intuitive-quaternions/

Re: The Octonion Math That Could Underpin Physics

#123

Post by NotPhysicsPhd »

PurpleBoxDragon wrote: post #118
sdenton4 wrote: post #13

I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "There are exactly four normed division algebras: the real numbers ($\R$), complex numbers ($\C$), quaternions ($\H$), and octonions ($\O$). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still…

Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a com…

Not sure about octonions. But quaternions are somewhat used in mechanics, specifically for dealing with rotations. The usage of quaternions is computationally simpler for describing arbitrary rotations in 3d dimensions.

Re: The Octonion Math That Could Underpin Physics

#124

Post by yaks_hairbrush »

PurpleBoxDragon wrote: post #118
sdenton4 wrote: post #13

I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "There are exactly four normed division algebras: the real numbers ($\R$), complex numbers ($\C$), quaternions ($\H$), and octonions ($\O$). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still…

Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a com…

Historically, quaternions came about as a way to try to reason about three dimensional physics. I mean, complex numbers were obviously really nice -- two dimensional numbers you could meaningfully add, subtract, multiply and divide. But they were only 2-D and we live in a 3-D world and we want to do 3-D physics.

So Hamilton was trying really hard to find a way to have 3-dimensional numbers that behaved nicely, and he couldn't do it. But he did find 4-dimensional numbers -- the quaternions. And they were really neat. You can use them quite well for classical mechanics, and electromagnetism, and even special relativity. So, why don't we?

We look at Maxwell's equations of electromagnetism today, and they're really nice, single-line vector formulas. You can also write them as nice, single-line quaternion formulas. Our notion of vector didn't exist at the time the quaternions were first used, and it was a boon to have quaternion notation to simplify some of these physical laws. Vectors and quaternions competed for a bit, and vectors won since they generalize to arbitrary dimensions.

Hidden inside of quaternion multiplication, you can find the three-dimensional versions of the dot product and cross product. And they do have some theoretically interesting properties for number theory and abstract algebra. In the end, however, sometimes items are discarded in favor of better items. I'd rate quaternions as one of the coolest items that ultimately wound up in the discard pile.

Re: The Octonion Math That Could Underpin Physics

#125

Post by daze42 »

PurpleBoxDragon wrote: post #118
sdenton4 wrote: post #13

I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "There are exactly four normed division algebras: the real numbers ($\R$), complex numbers ($\C$), quaternions ($\H$), and octonions ($\O$). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still…

Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a com…

I've known about quaternions for many years and even used them to program rotation matrices for a few 3d projects but I've never really given them much thought beyond that. For some reason, your comment just completely changed the way I thought about them. To think that the i part of ijk is the same i = sqrt(-1) is mindblowing to me. I had never considered that there may be other "more imaginary" dimensions that were required to solve yet more complicated problems.

Re: The Octonion Math That Could Underpin Physics

#126

Post by OscarCunningham »

bonzini wrote: post #103
gowld wrote: post #94

Earlier quoted context omitted.

Are there 32-onions that lack power associativity? https://en.wikipedia.org/wiki/Power_associativity https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...

No, power associativity is never lost, after sedenions the properties remain the same.

They don't remain exactly the same. The answers to this (https://math.stackexchange.com/questions/641809/what-specifi...) question provide some interesting starting-points.

Re: The Octonion Math That Could Underpin Physics

#127

Post by Iwan-Zotow »

anaphylactic wrote: post #67
qubex wrote: post #42

Earlier quoted context omitted.

That happens in general for matrices too.

Yes, and this is a bit obvious, but reals, complex numbers, split complex numbers, quaternions, octonions, sedenions, can all be represented as matrices of the appropriate form.

can't be true, unless there is some special matrix multiplication rule - afaik, standard matrix multiplication is associative

Re: The Octonion Math That Could Underpin Physics

#128

Post by logfromblammo »

PurpleBoxDragon wrote: post #118
sdenton4 wrote: post #13

I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "There are exactly four normed division algebras: the real numbers ($\R$), complex numbers ($\C$), quaternions ($\H$), and octonions ($\O$). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still…

Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a com…

If you code a system with 3D rotations coded as Euler angles, you can align two axises and lose a degree of freedom. This is gimbal lock.

Using unit quaternions (aka versors) to code the rotation instead, you cannot lose a degree of freedom. 4x4 matrices also solve this problem, but quaternions are more mathematically efficient. They also give very smooth interpolation for computer-assisted animations.

Also related to spinors in quantum physics and exists as a subalgebra of some conformal geometric algebras.

Re: The Octonion Math That Could Underpin Physics

#129

Post by joppy »

stochastic_monk wrote: post #9
adamnemecek wrote: post #6

I’ve been exploring this idea as well however I have a hunch it’s not actually octonions but dual quaternions as they are the perfect formalism for representing 3d movement over time. And to add to that, the are a Lie group I.e. they are anticommutative I.e. AB=-BA. I’ve also been exploring this relationship between dual quaternions and linear logic. It’s pretty wild. I’m curious if anyone has any opinions on this.

Do you have an understanding of Clifford algebras? I’ve read a little about them here on hn, and they seem quite powerful. I don’t understand them enough to know if they could also be an appropriate abstraction.

There is a process where you put in a space (number of dimensions) and a metric (Euclidean, for example) and produce a Clifford algebra. This is also sometimes called “geometric algebra”, although there are a lot of sensationalist posts using this name on the internet.

This Clifford algebra has an odd and even part, and the even part is an algebra in its own right, and is very useful for representing “rotations” in the space.

With the Euclidean metric, with one input dimension, the even Clifford algebra is just the reals. With two dimensions, you get the complex numbers. With three, the quaternions. However, since a Clifford algebra is always associative, you don’t get the octonions in this way.

Finally, if you put in four dimensions and the minkowski metric, the even Clifford algebra construction gives you a lovely algebra representing spatial rotations in three of the dimensions, and “boosts” along the time direction - exactly what you need to do calculations in special relativity.

Re: The Octonion Math That Could Underpin Physics

#130

Post by pnutz »

chunky1994 wrote: post #64

It's rather amusing that the author assumes that non-associtiave objects are "weird" for physicists (or at least that was my reading), since the velocity addition formula is in general non-associative and that has been extensively studied. (I remember three separate occasions in my undergrad particle physics class where we actually went through all the calculations involved with the velocity addition formula and fina…

Under what conditions is velocity addition not associative? In 1d they are associative right, but cant think in 3d
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