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The Peculiar Math That Could Underlie the Laws of Nature (quantamagazine.org)
xanthos writes: A fascinating article in Quanta magazine introduces us to Cohl Furey and the eight dimensional mathematics called octonions that she is using to model the interactions of strong and electromagnetic forces.
"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. The first three of these "division algebras" would soon lay the mathematical foundation for 20th-century physics, with real numbers appearing ubiquitously, complex numbers providing the math of quantum mechanics, and quaternions underlying Albert Einstein's special theory of relativity. This has led many researchers to wonder about the last and least-understood division algebra. Might the octonions hold secrets of the universe?"
"In her most recent published paper she consolidated several findings to construct the full Standard Model symmetry group for a single generation of particles, with the math producing the correct array of electric charges and other attributes for an electron, neutrino, three up quarks, three down quarks and their anti-particles. The math also suggests a reason why electric charge is quantized in discrete units -- essentially, because whole numbers are."
"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. The first three of these "division algebras" would soon lay the mathematical foundation for 20th-century physics, with real numbers appearing ubiquitously, complex numbers providing the math of quantum mechanics, and quaternions underlying Albert Einstein's special theory of relativity. This has led many researchers to wonder about the last and least-understood division algebra. Might the octonions hold secrets of the universe?"
"In her most recent published paper she consolidated several findings to construct the full Standard Model symmetry group for a single generation of particles, with the math producing the correct array of electric charges and other attributes for an electron, neutrino, three up quarks, three down quarks and their anti-particles. The math also suggests a reason why electric charge is quantized in discrete units -- essentially, because whole numbers are."
When cavemen see things they don't understand they invent a new god.
When physicists see things they don't understand they invent a new particle.
When mathematicians see things they don't understand they invent a new algebra.
It is left as an exercise for the reader to decide what Trump does when he sees something he doesn't understand.
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
Thats the only thing I got out of this headline
that women is a nerd
nah, normally 4 vectors are used which are NOT quaternions. Not seeing what advantage their use would give over four-vectors since they wouldn't represent space-time but rather space and operations in space.
Who ever put together that diagram about "Four Special Number Systems" was completely clueless about Mathematical Singularities
*facepalm*
NO, you do not. 0/0 is a singularity because it does NOT produce another real number. You get TWO numbers: +Infnity, and -Infinity and thus Mathematicians say the operation is "undefined".
"The math also suggests a reason why electric charge is quantized in discrete units — essentially, because whole numbers are"
So... you're telling me that reality is defined by an abstract algebra concept?
I thought we were using abstract algebras to *model reality*--not the other way around.
Comment removed based on user account deletion
When Computer Scientists sees a language they don't understand they invent a new "simpler" language.
When Chuck Norris sees something he doesn't understand he stares it down till he gets the information he wants.
Trump does when he sees something he doesn't understand, Sarah Huckabee says he did understand it.
Some drink at the fountain of knowledge. Others just gargle.
Nice thing about posting this story on Slashdot, is the ONE person that understands it, has left for greener pastures.
Some fried mushrooms to.
Everything is connected in ways we can't even comprehend.
The Russians have won. They have made the world a cesspool of distrust, greed, fear and hate.
All of a sudden the world is filled with women geniuses in every area. The past has been rewritten to include "very important" discoveries made by previously unknown women.
There is nothing in this garbage that hasn't been figured out by men long time ago. I'm sick with this propaganda. Fuck women!
So, the only thing wrong with the Timecube was that it was only half the story. Timecube is dead. Long live Octotime.
Don't disappoint your bird dog. Go to the range.
Great article and illustrates how as we try to understand reality (for lack of a better word): we first find that our current level of physics can't explain what we observe so we need to go to the next level. That next level needs the appropriate mathematical tools which often end up being already invented and looking for a practical application.
From the perspective of using a branch of mathematics that is new to the field, there's a lot of similarity between this story and using mathematics to predict crime: https://science.slashdot.org/s...
I believe we need to promote and retell these stories to students so that they can look beyond the simple and search for mathematical analogues that allow them to understand and model the physical world in different ways.
Mimetics Inc. Twitter
Can any of you smart mathematicians and physicists possibly down-translate this for the rest of us?
I'm sure I'm not alone in admitting I have not the slightest idea what the hell this is. OK, maybe I'm alone in admitting it, but I'm sure I'm not alone in having no idea what this is saying.
Without having to understand the physics or worry if it's right or not there is an important fact to be gleaned for computer scientists here. Specifically, we won't have a strong need to ever build SIMD systems wider than 8 (well maybe 16). There might be advantages for parallelism beyond that but they are merely scaling advantages not representational advantages.
That is to say, we currently handle 4 wide floats efficiently in SIMD systems. That's not an accident. Systems like Silicon Graphics were specially designed for exactly the purpose of efficient 4x4 matrix multiplication to handle quaternion graphics. Four is the essential number needed to make the atomic unit of all those transactions be the quaternion size. It makes everything else easier if you are not having to do bookkeepping on the data representation of the 4-vectors.
One might have thought that well, make an 8 then someone will want a 16 then a 32. So there's nothing special about 8. But this says indeed there is something special about 8. It's the largest size you really need to worry about the bookkeeping on. It's the largest atomic unit most algebras will ever need to treat.
You could scale beyond that but you will want to make sure that the most efficient ops can work on 8-vectors in whatever designs you consider in the future. it's special.
And microcode desginers will also want to make 8-ops special as well. Page boundaries should be multiples of 32= (8*float) etc...
Some drink at the fountain of knowledge. Others just gargle.
The article seems to concentrate on this one lady, and a bunch of what looks like pure mathematics. I'm interested in the actual physics, not a mathematical formalism.
A formalism, if you're curious is just a precise definition of a theory. There can can more than one equivalent formalisms for the same theory. Quantum Mechanics has 2 or 3 I believe, and they all predict the same thing. Think of it like polar co-ordinates vs Cartesian. Each is a tool where sometimes it's better to use one, and other times the other.
So what I'm curious about is... does this actually predict anything new, or is this just a new, innovative set of math to look at the same problem? That's really what the article SHOULD have done, but instead they just took a lot of artsy pictures of the lady, and some abstract math. If it's supposed to be about physics, where's the physics?
Sextonions are for me.
Their applications are extended from DNA to celestial bodies.
The really amusing thing to me is that historically, James Clerk Maxwell’s mathematical theory of electromagnetism (published in 1865), which for the first time unified electricity and magnetism, was written in the form of quaternions. For this reason, it was viewed by the engineering world as obtuse and impenetrable – 20 equations in 20 unknowns! Little was done with it until Oliver Heaviside re-wrote the theory in 1884 using the curl and divergence concepts of vector calculus, replacing 12 of the 20 equations with four short differential equations. Ironically, these four equations are now taught to undergraduates as “Maxwell’s Equations,” even though Maxwell never saw them (he died in 1879).
I’ve never seen an electromagnetics textbook written after 1900 that uses the original quaternion description of electromagnetics – they all use Heaviside’s vector calculus approach. It would be supremely ironic if a distaste of quaternions set the search for Physics’ Unified Field Theory back 150 years.
You're assuming horizontal SIMD, and ignoring vertical SIMD. Horizontal SIMD places values in the SIMD lanes corresponding to dimensions 'x', 'y', 'z', etc. Vertical SIMD places values in lanes corresponding to the same dimension across different items: e.g. 'x0', 'x1', x2', ....
The former is arguably bounded to a small finite number, the latter isn't.
Read the transcript: octo neons!
Or algebraic numbers?
"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.
If you read that kind of complete bullshit, it makes it hard to believe that any insights can be gained from reading the rest of the article. It's unlikely that the actually interesting stuff will have survived the apparently unreviewed incompetence of the author.
Octmushrooms!
It seems to imply that she is going through the process of refining the Standard Model and Quantum Mechanics in octonion maths, but isn't fully there yet. Some of it is done, but there is more to go. Perhaps once done, predictions will be made. Although it's does already seem to predict that there is nothing beyond the Standard Model, with a magic number being 8.
Octonion?
Is that a satirical "news" site for cephalopods?
... I was gonna say that.
I'm reminded of Oliver Heaviside, who refactored Maxwell's equations into the useful and familiar vector notation that has adorned many tshirts of electrical engineering and physics students. Heaviside took an unwieldy set of twenty field equations, and reduced them to four. I do wonder what insights we can potentially learn if the model itself is refactored into an elegant form.
Her PhD thesis: https://arxiv.org/pdf/1611.091...
The mathematician John Baez has an engaging writing style, and gave an amusing account of octonian numbers (His blog is very interesting BTW): http://math.ucr.edu/home/baez/
"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 respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."
http://math.ucr.edu/home/baez/...
Says that the reals, complex numbers, quaternions, and octonions are the only kinds of numbers that can be added, subtracted, multiplied, and divided. This is obviously false, as that can be done in any division algebra (including any field, like finite fields, rational numbers, etc, and there are there are uncountably many fields).
What they meant to say is that those are the only normed division algebras - basically, algebras over the real numbers with a notion of distance and such that the distance is compatible with multiplication.
"What lies behind us, and what lies before us are tiny matters compared to what lies within us." Ralph Waldo Emerson
So in the 50s a mathematician named David Hestenes developed a new branch of math called Geometric Algebra (based upon Clifford Algebras) which could subsume all of the different algebras used by physicists (and many others too). Additionally, it can handle contravariance and covariance, any positive integer number of dimensions, and handle algebras over imaginary numbers. Quantum Loop Gravity uses Geometric Algebra for instance. The problem is that Geometric Algebra isn't taught yet except perhaps at a post-doc level to mathematicians. The first textbook covering GA for Computer Science was just published in 2017. There are hopes that reformulating physics in to GA will allow unifications that were either not possible or too difficult when each part of physics uses different types of algebras.
The problem with all of this? GA is really really really hard. There is even an extension to GA called Geometric Calculus that's even more difficult. Given how difficult most students find VA which is much easier than GA, I'm not sure when we can expect most physics to make new theories using GA instead of VA. But when we can climb that hill, we will likely be able to see new physics on the other side. There are also a great many CS applications of GA as well (which is what I do).
My take on TFA, is that this physicist is going down a wrong path because she was never taught GA. If she finds something, it will likely have to be converted into GA to unify it with other algebras used in other parts of physics. But I could be wrong, who knows but some of the greatest physicists in history have gone down this specific rabbit hole with nothing to show for it at the end. I wish her luck.
"Those that start by burning books, will end by burning men."
Can any of you smart mathematicians and physicists possibly down-translate this for the rest of us?
I'm sure I'm not alone in admitting I have not the slightest idea what the hell this is. OK, maybe I'm alone in admitting it, but I'm sure I'm not alone in having no idea what this is saying.
Around 1940 (IIRC), Eugene Wigner pointed out that symmetries in physics let is map physical theories to abstract groups, and this can place restrictions on what the correct equations have to be, in a way that lets us winnow down the possible theories to only those that satisfy the group topologies.
Suppose you have a square playing card nestled in a square indentation on a table (a regular playing card, except it's square instead of rectangular). How many ways are there to pick up the card and place it back down in the indentation?
The answer is 8 possible ways. If you paint one of the edges of the card, then there are 4 possible sides (of the hole) where the painted edge can go, and then you can have the card face-up or face-down. Each of these placements corresponds to a rotation or a flip of the card: Four rotations (including the identity rotation of 0 degrees), and four flips, along the vertical, horizontal, or two diagonal axes.
No matter how many rotations and flips you make, you always end up in one of the 8 basic positions. Thus, the operations form a group - called the "dihedral" group. The operations are closed: no matter how many flips and rotates you use, it ends up as the same as one of the original 8. Each operation has an inverse, and the 0 degree rotation acts as an identity element. (It's also associative, but that's difficult to show.)
Now imagine the card centered on the X-Y plane, and draw 4 vectors from the origin out to each of the four corners. You can define 8 matrices that flip the vectors in various ways, each matrix being associated with one of the flip or rotate operations.
Thus, the 8 matrices become a representation of the dihedral group. This puts some strong restrictions on the types of matrix you use: each matrix has to have length 1 (it can't change the length of the vectors), and you can't flip one edge over without flipping the opposite edge, because you can't "twist" the card. The matrix length can't be -1 because that would make the card a mirror image - the "J" of a Jack would curve to the right instead of the left.
You can now use matrix mathematics to prove things about your group.
For a different group, consider a vector going from the origin to the unit sphere. You can consider all matrices that rotate the vector in 3D without changing its length or moving its origin. This also forms a group (operations are closed, operations have inverses, and there's an identity operation), but it's an infinite group (a Lie group) and the sphere surface is "smooth". This means that you now can now use differential geometry to prove things about your group.
This group is called SU(3), the "Special Unitary group". It's "Unitary" because the rotations don't change the lengths of the vectors (the matrices are of length 1), and it's "Special" because it doesn't allow mirror-images: the determinant ("length") of the matrix cannot be -1, in the same way that we can't have a matrix of length -1 when rotating cards.
Now consider a physics experiment. We set up an apparatus, calculate the wave equation, and at the end we measure (for example) the energy. We measure energy by applying an operator to the wave equation that describes the experiment.
We can imagine rotating our point of view around the experiment, so that when we do the experiment we measure the energy looking from the other side of the apparatus.
We expect in that case to get the same value.
This means that the energy operator we apply to the wave eq
I get the role of quaternions and octonions. What I don't get is the role of the other onions.
I mean, who is looking into the scallions, the chive, the shallots, the endive? What about leeks and garlic? Wake up physisheeple! We're never gonna crack the TOE, dark whatchamacallits, and the infamous toast flipping problem at this rate!
Geez...
Her name sounds like a comic book super hero. Cold Fury?
well yes I think that's what I said in other terms. the representational format (as you called "x y z t") verus the pures scaling format (0,1,2,3,4..). the limit of 8 on the divisional algebras makes this special for the representational one.
Some drink at the fountain of knowledge. Others just gargle.
I'm placing a Vagina Hex on you
I'm giving you Cooties.
I'm passing word to all the women to not go out with you
also, your penis is small and I'm laughing at you in front of everyone.
Now go back to your hate porn and squeeze one off.
and if that is proven true... that is an amazing feat in itself.
TLDR: In principle you are always using the reals.
Oh, also FTFY
You get only one ring, but it's really tangled up with itself
Currently, it doesn't predict anything. HOWEVER, once complete (if successful), it will be a cleaner and more natural formalism. It should also be able to unify relativity/gravity with QM. THAT should produce some strong predictions.
Remember Mathematics only MODELS reality. It isn't reality itself. It is just a model of reality we can understand and manipulate.
People forget this about science too. Science does not "prove", it only describes.
as he felt they had for him.
This would add weight to the argument I first read 10 years ago that universities have become more and more narrative focussed instead of practical focused in their funding.
Pythagoras called, he wants his idea back
Excuse me sir do you have any spare brain cells.
This is still above my pay grade.