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The Octonion Math That Could Underpin Physics

quantamagazine.org

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Re: The Octonion Math That Could Underpin Physics

#41

Post by adamnemecek »

electricslpnsld wrote: post #38
adamnemecek wrote: post #10

Earlier quoted context omitted.

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.

> http://www.chinedufn.com/dual-quaternion-shader-explained/ Offset curve deformations blow dual quaternion's out of the water for rigging!

Tell me more. I briefly looked it up and I do agree it does look somewhat nicer. However how's the performance.

Re: The Octonion Math That Could Underpin Physics

#42

Post by qubex »

Maybestring wrote: post #7
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…

>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

That happens in general for matrices too.

Re: The Octonion Math That Could Underpin Physics

#43

Post by tlb »

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?

That's a big topic, and the definition of simple is subtle. I recommend David Deutsch's book _The Beginning of Infinity_ for an accessible introduction to what makes one theory better than another when they both fit the observations.

Re: The Octonion Math That Could Underpin Physics

#44

Post by arketyp »

_bxg1 wrote: post #15
ianai wrote: post #2

I 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.”

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.

Gödel was motivated by convictions about meta-mathematics, motivated by his Catholic fatih. Perhaps a purely mechanistic world view would have eventually generated the same result, e.g. Turing's halting problem, but who knows.

Re: The Octonion Math That Could Underpin Physics

#46

Post by Steuard »

jstanley wrote: post #27
nabla9 wrote: post #18

Earlier quoted context omitted.

Being composition algebras makes R,C,Q and O more interesting than others: N(xy) = N(x)N(y) - N is called norm. Without this property, you have zero divisors. edit: throwawaymath uses better notation: |xy| = |x| • |y|

What type of thing is N? What you wrote doesn't seem to make sense if N is just a constant, but I don't see how it makes sense if N is a function? Or maybe you didn't mean multiplication? E.g. N = 3, x = 2, y = 3 N(xy) = 3(2 . 3) = 18 N(x)N(y) - N = 3(2) . 3(3) - 3 = 51, which is not 18

N here is a function, not a constant. I was confused at first, but I think the "-N" in that post wasn't intended to mean "minus N" but rather ", where N is the function called...".

The "norm" of a number is more or less its absolute value: its size, it's magnitude. So "1" has norm 1, but "-1" also has norm 1, as does "i", and "-5" or "5i" or "4-3i" all have norm 5. So these four mathematical structure (R,C,Q, and O) all have the property that the norm of a product is equal to the product of the norms. Things get really obnoxious (or at least really unfamiliar) if you don't have that property.

Edit: Sorry for the repetition! I've got to remember to reload these pages before replying.

Re: The Octonion Math That Could Underpin Physics

#47

Post by Sharlin »

throwawaymath wrote: post #16
paulddraper wrote: post #12

Earlier quoted context omitted.

Example: rational numbers

The rationals are a subfield of the reals.

Yeah, but they are still a counterexample to the stated claim. It could be that only ℝ as a whole forms a field.

Re: The Octonion Math That Could Underpin Physics

#48

Post by kjeetgill »

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?

I think there's a little more foundation to it that just "Occam said it". I think the point is that the simplest theory is a) a lower bound and b) most valuably falsifiable.

Take a theory, add a bit about a Unicorn. or a Dragon. Now you have a more complicated theory. You can arbitrarily contrive a more complicated solution.

But the simplest? You've found a local minima. And because it's more stable you can compare it to competing theories without trivial refutation. see: the 3-5 theories of dark matter.

Re: The Octonion Math That Could Underpin Physics

#49

Post by whatshisface »

tlb wrote: post #43
whatshisface wrote: post #35

Earlier quoted context omitted.

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

That's a big topic, and the definition of simple is subtle. I recommend David Deutsch's book _The Beginning of Infinity_ for an accessible introduction to what makes one theory better than another when they both fit the observations.

One theory might be better than another when they both fit the observations, but you can't say which one is true-er. For example, were Newton been presented with GR but only allowed his contemporary evidence, he should not conclude that classical gravitation is more true than GR on the grounds that he doesn't have any evidence for the additional complexity.

Re: The Octonion Math That Could Underpin Physics

#50

Post by btrettel »

ianai wrote: post #36
madhadron wrote: post #24

Earlier quoted context omitted.

This is very false. Einstein's reputation was built on explaining known phenomena such as the photoelectric effect and Brownian motion. Special relativity was heavily motivated by a pile of puzzling evidence and a bunch of existing mathematics.

And he funded his studies, initially, as a hobby alongside a job. We’ve not progressed very far if ground breaking research almost requires a person to find a novel study path.

What do you mean by "novel study path" here? Funding yourself with money received from a job?

On my first reading I took "novel study path" to mean new research techniques or something along those lines, which seems obvious and is not likely what you meant.

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