Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Octonion Math That Could Underpin Physics

quantamagazine.org

Post reply on HN
Print view View original

151–160 of 192 posts

Re: The Octonion Math That Could Underpin Physics

#151

Post by tedajax »

XorNot wrote: post #143
scotty79 wrote: post #135

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.

I've always found the five perfect solids kind of interesting for that reason. Seems so weird for the set to be so finite.

Re: The Octonion Math That Could Underpin Physics

#152

Post by marlag »

> They “imagined that the next bit of progress will come from some new pieces being dropped onto the table, [rather than] from thinking harder about the pieces we already have,”

When the best minds we have divide into two camps, one saying that (a) "with X and Y we don't have enough information to solve Z" and the other saying (b) we do but we need to think harder, the first camp builds a particle collider, the other creates what, string theory? Aren't we doing "fuzzy science" here?

It seems that the experts in one of the camps should go back and retrace their steps because somewhere along the line they made an assumption based on some data that they (I assume) forgot to encode into their equations and now they have trouble taking it to the next level. Why is it not clear to us that eiher a or b is true?

Re: The Octonion Math That Could Underpin Physics

#154

Post by sannee »

danbruc wrote: post #82
sannee wrote: post #76

Earlier quoted context omitted.

> In the approach from the article it seems as you are picking a mathematical structure in the middle of nowhere Quaternions were very popular way of expressing the "classical" physics around the 19th century (and the vector algebra we know today is in some ways just a derivative of quaternion algebra). Complex numbers are extremely useful in many fields even today. It's hardly in the middle of nowhere.

But the article is not really talking about quaternions - which are surly a useful tool, probably best known for the nice way in which they can describe rotations - but about R⊗C⊗H⊗O. And it's at the very least not obvious that this thing is anywhere close to where they journey is hopped to lead to.

Octonions are to quaternions as are quaternions to complex numbers (and complex numbers to reals). This is called "Cayley construction" iirc.

Re: The Octonion Math That Could Underpin Physics

#155

Post by SubiculumCode »

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.

An n-dimensional matrix of octonions

Re: The Octonion Math That Could Underpin Physics

#156

Post by kbwt »

XorNot wrote: post #142
yaks_hairbrush wrote: post #124

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…

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.

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.

Re: The Octonion Math That Could Underpin Physics

#157

Post by chombier »

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.

one of the most intriguing/funny things about dual numbers (and dual quaternions in particular) is how they encode the derivative of the real part into the dual part, and how everything plays out nicely algebraically.

this could be a nice implementation trick for an automatic differentiation framework.

Re: The Octonion Math That Could Underpin Physics

#158

Post by yoklov »

kbwt wrote: post #156
XorNot wrote: post #142

Earlier quoted context omitted.

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.

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 extremely fast.

Re: The Octonion Math That Could Underpin Physics

#159

Post by danbruc »

sannee wrote: post #154
danbruc wrote: post #82

Earlier quoted context omitted.

But the article is not really talking about quaternions - which are surly a useful tool, probably best known for the nice way in which they can describe rotations - but about R⊗C⊗H⊗O. And it's at the very least not obvious that this thing is anywhere close to where they journey is hopped to lead to.

Octonions are to quaternions as are quaternions to complex numbers (and complex numbers to reals). This is called "Cayley construction" iirc.

Sure, but the structure the article is about is the tensor product of the four algebras which is an algebra with 64 real dimensions.

Re: The Octonion Math That Could Underpin Physics

#160

Post by crocal »

romwell wrote: post #144
donbright wrote: post #137

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…

This reasonates with me so much. Thanks for the tip.
Post reply on HN