Can anyone expand on the current successful similarities between octions and the standard model in plain English?
The Octonion Math That Could Underpin Physics
141–150 of 192 posts
Re: The Octonion Math That Could Underpin Physics
#142Earlier quoted context omitted.
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…
Quaternions show up all the time in 3d graphics - they're how you represent rotation matrices without the problem of gimbal lock.
I don't know if that's significant but it was how I stumbled on the concept in high school when I was messing with Direct X.
Re: The Octonion Math That Could Underpin Physics
#143Earlier quoted context omitted.
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…
Complex numbers tell you what happens if there is an i that i^2=-1. What happens is that you get cool way to express 2d rotations. Quaternions tell you what happens if there are three different i's that have this property. And it tells that it leads to nice algebra that expresses 3d rotations very well (and even 4d if you believe the article). Octonions tell what happens if there are seven such i's. And it leads to c…
I think the main curiosity stems from the fact that octonions are as far as you can go.
In math infinity turns up all the time so having there be exactly a finite number of anything feels weird.
Re: The Octonion Math That Could Underpin Physics
#144Earlier quoted context omitted.
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 did not understand quaternions until I read Hamilton's original works. Maybe i just have a 19th century brain or something. but i found them delightfully free of modern gobbldeygook. https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/
>Maybe i just have a 19th century brain or something. but i found them delightfully free of modern gobbldeygook.
No, you can blame Bourbaki for that. People such as V. Arnold decried the way mathematics is now presented[1].
It was from Hamilton's book that I learned what the word vector means and why it's used. It simply means carrier (as in malaria vector that you might heard from biologists) - and carries the space, by a translation!
Such lucidity is absent from all linear algebra books I've seen.
We need to go back to the presentation style of 19th century, where not only the result, but the thought process is presented. Today's papers look like they are written for formal verification systems.
[1]https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
Re: The Octonion Math That Could Underpin Physics
#145> 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…
is there a proof that real numbers can be added?
As far as I know, addition is something you define (an axiom) in order to construct the real (or other) numbers.
Re: The Octonion Math That Could Underpin Physics
#146Earlier quoted context omitted.
It's my understanding that most of Einstein's theory was the product of intuition, backed up after-the-fact with mathematics and experimentation. Intuition isn't a bad compass, as long as you can set it aside if reality measurably contradicts it. In fairness, Einstein never accepted quantum mechanics because they flew in the face of his intuition, but it still got him pretty far.
> Einstein never accepted quantum mechanics Wait? Since when did Einstein not accept Quantum Mechanics? He won the Nobel Prize for his work in discovering an important part of Quantum Mechanics. Einstein rejected the Copenhagen Interpretation of Quantum Mechanics. That's not the same thing as rejecting Quantum Mechanics. I've seen no claims that he would object to either the Everett or Bohm Interpretations of QM. (Bu…
Well, he never accepted non-locality. And it turns out, quantum mechanics is indeed non-local.
Re: The Octonion Math That Could Underpin Physics
#147Earlier quoted context omitted.
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…
It even better ii = jj = kk = ijk = -1
You also have ij = k, jk = i, ki = j
What are these thing? I have no idea, nor did I know there are 4 more - and only 4 more.
Re: The Octonion Math That Could Underpin Physics
#148Earlier quoted context omitted.
>Is there an easy explanation of what problems quaternions solve? Sure. Unit quaternions form a double-cover of SO(3). In other words, you can encode a rotation of a 3-dimensional object with a single unit quaternion. But wait, there's more! You could do the same with a matrix, or a triple of angles. Why not do that? Answer: interpolation. The "natural" way you want to go from one rotation to another corresponds to e…
In my opinion this is the best answer, because it neatly explains why quaternions are more useful than vectors for rotation. Vectors are "nicer" because they generalize to arbitrary dimensions. But quaternions handle 3-dimensional rotations in (essentially) a single step. The point about interpolation is really important, because all the machinery offered by vectors becomes a burden. It's also good to think about com…
Vectors are still in play IIRC; quaternion rotations are an application of the rotation theorem.
Re: The Octonion Math That Could Underpin Physics
#149Earlier quoted context omitted.
Complex numbers tell you what happens if there is an i that i^2=-1. What happens is that you get cool way to express 2d rotations. Quaternions tell you what happens if there are three different i's that have this property. And it tells that it leads to nice algebra that expresses 3d rotations very well (and even 4d if you believe the article). Octonions tell what happens if there are seven such i's. And it leads to c…
I think the main curiosity stems from the fact that octonions are as far as you can go. In math infinity turns up all the time so having there be exactly a finite number of anything feels weird.
You can do sedenions, they lose another property and beyond that they aren't interesting because no properties really change...
Re: The Octonion Math That Could Underpin Physics
#150I have to disagree that quaternions underlie Special Relativity. Although special relativity does use 4-vectors, those aren't quaternions.
I remember going down that path when I was an undergraduate. Fortunately I had a very experienced theorist to hand who explained that, yes, people did try that, but stopped bothering because it doesn't generalize to general relativity, and there was no point keeping two mathematical toolboxes around when you could have one.
Clifford algebras are great for both SR and GR and have all the properties that one likes in quaternions (and more)