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The Octonion Math That Could Underpin Physics

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Re: The Octonion Math That Could Underpin Physics

#131

Post by logfromblammo »

yaks_hairbrush wrote: post #124
PurpleBoxDragon wrote: post #118

Earlier 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…

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 have a similar relationship with a 4-D geometric algebra, where one dimension is a timelike dimension, because that is one gnarly mess of anticommutative, nonassociative math to wade through.

Re: The Octonion Math That Could Underpin Physics

#132

Post by HelloNurse »

whatshisface wrote: post #35
OscarCunningham wrote: post #19

Earlier quoted context omitted.

Surely what we're looking for is the simplest theory among the theories that hit all the datapoints. That's the theory that's most likely to be the "Theory of Everything" and therefore to continue working when we upgrade our collider.

Why would the simplest theory be the most likely to be true?

Scientific theories are not likely to be true: they are either true enough to be taken seriously or false enough to be dead. Accumulation of evidence and knowledge makes scientific theories descend a ladder of wrongness that goes from unthinkable, to utterly ridiculous, to wrong but respectably clever, to somewhat grounded in reality, to good enough approximations for some purposes, to best in class but not perfect, to positively agreeing with all available evidence.

Given equally true theories scientists look to other properties to establish theory quality, and simplicity is a philosophically important one: it represents the belief that natural laws should be as simple and elegant as evidence allows them to be.

Re: The Octonion Math That Could Underpin Physics

#133

Post by sleepyams »

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…

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...

IDK much about octonions, but quaternions have similar algebraic properties to rotation groups and can be used to encode rotations in 3D graphics.

Re: The Octonion Math That Could Underpin Physics

#134

Post by sleepyams »

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…

In addition to the other replies, check out Clifford algebras which generalize these types of numbers:

https://en.wikipedia.org/wiki/Clifford_algebra

Re: The Octonion Math That Could Underpin Physics

#135

Post by scotty79 »

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…

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 cool algebra that helps expressing.... we don't quite know yet.

Also considering other numbers of i's doesn't lead to anything coherent. Also considering different possibilities of what should be the result of multiplying one i by another different i doesn't lead to anything cohesive.

So... i^2=-1 apparently can have either zero, one, three or seven solutions and they have to have very specific relationships between themselves for calculations to make sense

Curious...

Re: The Octonion Math That Could Underpin Physics

#136

Post by romwell »

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…

>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 exponentiation of quaternions. If you linearly interpolate matrices, the intermediate steps will do something nasty: they won't even be rotations!

The natural way to implement the Arcball interface for rotations is using quaternions. Here[1], I have implemented it in ProcessingJS and wrote up the math behind it.

Quaternions (like complex numbers) can do other things too, but this alone is a good start. Also gives you intuition why they aren't commutative: because rotations in 3-space aren't.

TL;DR: Unit complex numbers = rotations of plane. Unit quaternions = rotations of 3-space.

PS: you shouldn't think of complex numbers as the solution to the problem of "taking the square root of -1". Think of them as "how can I multiply/divide a 2D vector by another 2D vector?" - there's only one way to do it sanely (multiply/divide lengths, add/subtract angles). This is what the complex numbers are.

Hamilton was trying to solve the same problem in 3D, and couldn't (turns out, it is not possible[2]), but solved it in 4 dimensions, and later found many applications for them.

[1] http://www.math.tamu.edu/~romwell/arcball_js/index.html

[2] https://en.wikipedia.org/wiki/Frobenius_theorem_(real_divisi...

Re: The Octonion Math That Could Underpin Physics

#137

Post by donbright »

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 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/

Re: The Octonion Math That Could Underpin Physics

#138

Post by donbright »

anaphylactic wrote: post #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. Wit…

is there a proof that real numbers can be added?

Re: The Octonion Math That Could Underpin Physics

#139

Post by eggy »

joppy wrote: post #129
stochastic_monk wrote: post #9

Earlier quoted context omitted.

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 t…

I am working my way through Understanding Geometric Algebra by Kenichi Kanatani up to the Grassmann Algebra chapter, so I can then pick up John Browne's Grassmann Algebra Volume 1 Foundations.

Kenichi's book works from basic vector calculus, Hamilton, Grassmann and Clifford. Grassmann was not even a mathematician by trade, but a school teacher and a linguist. He translated the Rig Veda. His big mathematical work did not get the attention when written as it did posthumously. I know things with cool names like differntial forms and fiber bundle space [1]. I'm becoming quite the Grassmann fanboy!

I am thinking if I can learn these algebras (Grassmann, Clifford, etc...) I will be able to deal with more abstract thinking in geometry and handle it more concisely and logically. Almost like when I use J programming language for math! [2]

I think Quarternions don't hold after 3 dimensions, so what about Octonions?

  [1]  https://galileospendulum.org/2012/04/26/everything-is-geometrical-hermann-grassmanns-algebra/
  [2] jsoftware.com

Re: The Octonion Math That Could Underpin Physics

#140

Post by throwawaymath »

romwell wrote: post #136
PurpleBoxDragon wrote: post #118

Earlier 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…

>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 complex numbers in the way described here because it neatly abstracts the concept of numbers and maps them to a form of dimensionality. Great comment all around.

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