The Octonion Math That Could Underpin Physics
quantamagazine.org
The Octonion Math That Could Underpin Physics
1–10 of 192 posts
Re: The Octonion Math That Could Underpin Physics
#2I think the following quote is the QED for academia ruining any chance at actual research:
“What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”
Re: The Octonion Math That Could Underpin Physics
#3> There the game stops. Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided.
1. I think they meant "the only kinds of numbers constructed in this way". 2. Sedenions can still be added, multiplied, subtracted and divided. it's just that multiplication and division lose most of their useful properties. With octonions you've already lost associativity and commutativity, though.
Re: The Octonion Math That Could Underpin Physics
#4I think that maybe if you lock a mathematical physicist in a box forever with an x-TeV collider without letting them upgrade it, they will eventually find a theory that hits all of the datapoints and depending on their philosophical weakness then declare it "final."
Re: The Octonion Math That Could Underpin Physics
#5"Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided. "
Not true. Any field (in the algebra sense) has these properties. [1]
Quaternions and octnonions also have weirder properties: quaternions are non-commutatve (jk=-kj) and octnonions are non-associative: a(bc) != (ab)c.
I think the article meant these are the only Euclidean Hurwitz algebras [2], which is a far cry from the claim.
[1] https://en.wikipedia.org/wiki/Field_(mathematics) [2] https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...
Re: The Octonion Math That Could Underpin Physics
#6I’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.
Re: The Octonion Math That Could Underpin Physics
#7> There the game stops. Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided. 1. I think they meant "the only kinds of numbers constructed in this way ". 2. Sedenions can still be added, multiplied, subtracted and divided. it's just that multiplication and division lose most of their useful properties. Wit…
>it's just that multiplication and division lose most of their useful properties.
Specifically they lose the property of not having zero divisors.
There exists sedonions a,b != 0 such that ab = 0
Re: The Octonion Math That Could Underpin Physics
#8Her name is Cohl Furey, that is SO COOL!
Re: The Octonion Math That Could Underpin Physics
#9I’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.
Re: The Octonion Math That Could Underpin Physics
#10I’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.
Yes! Dual quaternions are a Cl(2,0,1) Clifford algebra. Good ol Cliffy invented them too.
http://www.chinedufn.com/dual-quaternion-shader-explained/
Check the guy on the right getting deformed like an idiot. Compare it with the smooth criminal on the right.