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The Octonion Math That Could Underpin Physics

quantamagazine.org

161–170 of 192 posts

Re: The Octonion Math That Could Underpin Physics

#161

Earlier quoted context omitted.

Less edge cases mean it's less likely to break if we discover a new particle or something else unexpected

I guess that's a good overview. It's related to the phenomenon of overfitting in machine learning: you can always easily find a sufficiently complex (or complicated, large if you prefer) theory fitting all data points. Because this theory simply encodes each observed case (including progressive sophistications of encoding), you naturally expect it to fail on unobserved cases -- it makes no effort at generalization. T…

> Honestly I haven't seen attempts at making this process more rigorous, when applied to physics.

Marcus Hutter has expressed this idea quite well ( https://arxiv.org/pdf/0912.5434 ) arguing that (a) smaller/simpler theories have more predictive power and (b) the "size" of a theory includes the complexity of its equations and the parameters needed to specify some result. The latter is important because some theories trade off between these two: e.g. a multiverse theory might have simple equations ("every possibility happens somewhere") but require very precise "coordinates" to pin-point the actual possibility that we observe.

Not sure if other physicists know of or take it seriously though.

Re: The Octonion Math That Could Underpin Physics

#163

If it goes: 1D, 2D, 4D, 8D, why does it stop at 8? Is there a hexadeconion?

They are called sedenions [1] and you can repeat the construction taking you one level up [2] infinitely.

[1] https://en.wikipedia.org/wiki/Sedenion

[2] https://en.wikipedia.org/wiki/Cayley–Dickson_construction

Re: The Octonion Math That Could Underpin Physics

#164

Earlier quoted context omitted.

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…

If you liked Hamilton, you should try Lanczos' Varational Principle of Mechanics.

Re: The Octonion Math That Could Underpin Physics

#165

So, did Furey get a faculty job or can she be viewed with her accordion somewhere?

FWIW, I'll certainly tell my department chair in New Orleans about her if they're doing another faculty search soon. Maybe she can have her cake and eat it too that way.

Re: The Octonion Math That Could Underpin Physics

#166

Earlier quoted context omitted.

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…

Geometric algebra, related to quaternions, can represent Maxwell's equations in a single equation, using polyvectors and the inner and outer products. The subalgebras formed by scalar and bivectors (1, x^y, y^z, z^x)--or the monovectors and the pseudoscalar (x, y, z, x^y^z)--from a three-dimensional geometric algebra have a lot of the same mathematical properties as quaternions. I don't know whether or not octonions…

You can summarize Maxwells equations in any dimension using two formulas: $$ \delta F = j d F = 0 $$ where $F$ is the field-strength two-form and $j$ is the current density.

Re: The Octonion Math That Could Underpin Physics

#167
post #143

Earlier 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.

There is an infinite number of them.

https://en.wikipedia.org/wiki/Cayley–Dickson_construction

They're nuts, though.

Re: The Octonion Math That Could Underpin Physics

#168
post #113

Earlier quoted context omitted.

And why are the real numbers not called "onions"?

Obviously due to potential confusion with the zero-dimensional numbering system, "0nions".

Hehe, well we could have unions & oneions. For some reason as I'm reading about octonions, I keep picturing 8-cloved garlic.

Re: The Octonion Math That Could Underpin Physics

#169
post #158
post #156

Earlier quoted context omitted.

Rotation matrices by themselves do not suffer from gimbal lock. I think you meant Euler angle representations with the rotations always applied in a consistent order around the pitch/yaw/roll axes.

Right. Just to add on, one reason Quaternions are still used in graphics (despite rotation matrices not suffering from gimbal lock either) is that they're easy to interpolate between, even if you have many. If you just have two you can slerp (or not), but if you have a large number of them (weights from an animation system, for example), a basic weighted sum followed by normalizing is shockingly well behaved and extr…

This is a timely comment. I've been aware of quaternions for rotation for decades but only learned yesterday that rotation matrices can be used for some use cases. I'd never heard this small fact until then and now I stumble across your comment!
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