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Math

The Peculiar Math That Could Underlie the Laws of Nature (quantamagazine.org) 242

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."

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The Peculiar Math That Could Underlie the Laws of Nature

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  • quanternions for SR? (Score:4, Informative)

    by iggymanz ( 596061 ) on Friday July 27, 2018 @11:57AM (#57019570)

    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.

    • by goombah99 ( 560566 ) on Friday July 27, 2018 @12:35PM (#57019846)

      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...

      • 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.
        • 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.

      • by mikael ( 484 )

        There are the AVX, AVX-256 and AVX-512 extensions to the x64 instruction set. 128-bit width gives four floating point, 512-bit width registers gives sixteen floating point values or eight double precision values.

    • by lgw ( 121541 )

      nah, normally 4 vectors are used which are NOT quaternions.

      You're thinking of General Relativity, not Quantum Mechanics. QM math is all about matrices of complex numbers..

      Quaternions and and matrices are somewhat annoying, as they're not commutative. Octonions are neither commutative nor distributive, which makes them harder still to work with.

      • wasn't thinking of QM at all but Special Relativity which uses 4 vectors.

      • by rtb61 ( 674572 )

        Things just get weird when you try to take free floating energy out of the system. No particle to contain the energy, there is no energy. It is all discrete particle flows and the energy contained within those particles, both the inherent energy of the particle and the transient energy as it transfers from particle to particle, balancing out. The infinitely small particles of quantum space (relative to normal space) and infinite large particles as the universe itself could be considered an individual partic

    • by BitterOak ( 537666 ) on Friday July 27, 2018 @01:35PM (#57020224)
      True, ordinary quaternions aren't that useful for describing spacetime but biquaternions [wikipedia.org] give a very natural and elegant way to model the space-time of special relativity. In particular, Maxwell's equations can be written as one simple equation which is manifestly covariant. Lorentz transformations in this algebra have the matrix representation SL(2,C), the set of complex 2x2 matrices with determinant one which is the covering group of the 4x4 matrix algebra representing proper, orthochronous Lorentz transformations. In a sense, biquaternions are to Lorentz transformations in special relativity what quaternions are to rotations in three dimensional Euclidean space.
      • by MrMr ( 219533 )
        I clicked on the link. Gave me the impression 'natural', 'elagant' and 'simple' may mean something different where you live.
    • by drakaan ( 688386 )
      ok, and I don't know the math well enough to feel I can speak intelligently, but...doesn't the whole article talk about octonions being the area of investigation. Which would not be quaternions, right?
      • by ceoyoyo ( 59147 )

        The article describes using real numbers (obviously), complex numbers (also fairly trivial), quaternions AND octonions. That's what the RxCxQxO thing is about.

  • Who ever put together that diagram [cloudfront.net] about "Four Special Number Systems" was completely clueless about Mathematical Singularities [wikipedia.org]

    When you add, subtract, multiply or divide the "real numbers" used in everyday life, you always get another real number

    *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".

    • by iggymanz ( 596061 ) on Friday July 27, 2018 @11:59AM (#57019584)

      + and - infinity aren't numbers, and no they really don't solve the 0 / 0 problem. that quotient is just undefined for useful maths

      • Thanks for the catch! I knew I missing something.

        I should have put air quotes around: You get TWO "numbers:" to signal that infinities are concepts and not numbers -- despite that they tend to be treated like numbers.

        Thanks again!

        • by mark-t ( 151149 )
          Actually, you don't get any number. Division by zero is entirely undefined and meaningless. It is only correct to say that the limit of division approaches plus or minus infinity as the divisor approaches zero. But the limit of a division by a number that approaches zero is not the same thing as division by zero. The former is actually defined, having two specific (non-numeric) values: +/- infinity. The latter is entirely undefined. It is about as meaningful (and no more correct) to say n/0 = +/- inf
      • by dgatwood ( 11270 )

        And yet nobody has trouble with square roots being defined, even though sqrt(4) is +/- 2.

    • by fisted ( 2295862 ) on Friday July 27, 2018 @12:30PM (#57019804)

      It's almost as if the diagram targets scientifically curious laypeople, so your nerd rage about this irrelevant detail (given the context) is a bit over the top.

      • Except the _exceptions_ themselves ARE interesting.

        They could have easily added a minor foot note:

        You always (*) get another real number.
        (*) Except division by zero

        Likewise the Complex number is crap for omitting the detail that i represents a 90 degree counter clockwise rotation.

        The defining characteristics of i is that it represents a 90 degree CCW; thus its square is negative. i.e. i^2 = -1

        --
        Mojang "Logic"

        You will not be able to ride dolphins that is animal cruelty.

        "Riding" digital pixels such as pig, ho

    • That's probably true for all 4 named systems.

      However, the article is written at a level for most people to gain some understanding. Adding in the exceptions would just needless complicate the article, and take away from the points he's making.

    • Actually you don't get infinity, because 0*infinity is undefined. 0/0 is undefined because it "produces" all numbers (i.e. you can pick any number, and it'll work). You need to rebuild the problem as a limit to see if it converges to any particular number.

      Interestingly, 0*infinity is undefined for the same reason.

    • by ceoyoyo ( 59147 )

      Mathematicians have the useful concept "almost everywhere" (https://en.wikipedia.org/wiki/Almost_everywhere) so they can talk about useful properties while remaining rigorous enough for the most determined pedants.

  • Everything is connected in ways we can't even comprehend.
  • So, the only thing wrong with the Timecube was that it was only half the story. Timecube is dead. Long live Octotime.
  • by mykepredko ( 40154 ) on Friday July 27, 2018 @12:26PM (#57019768) Homepage

    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... [slashdot.org]

    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.

    • Too bad mathematics has, after all this time, no feed-forward closed-form solution whatever to the N body gravitational problem.
      Much less one with charge (Coulomb) and strong force.
      Or for that matter, any recursive/fractal problem - weather is a simple example. You just have to perturb endlessly.
      We'd have fusion if they did.
      Now, trying to simulate a 10^20 something body problem as a particle in grid is heat death of the universe difficult.
      Math isn't the queen of all the science we need. It was just lu
      • by zlives ( 2009072 )

        again this would perhaps mean that we have been looking in the wrong place for the answer. that is what makes this story interesting to me, perhaps a lot of science energy has been wasted on colliders as answers when wee havn't yet gotten to the right math.

        • We could wish they'd fund more than one approach, to be sure. There's no one "wrong place" and I suspect, more than one useful place to find clues. I have a friend at CERN and we've discussed their detector filtering algos - they need to do some reduction in data rates. The thing is, at some point in this process, it's easy to eliminate anything you weren't looking for anyway...and they know it. Technical limitations as much as anything, but also cognitive.
          .

          "Expect the unexpected" is harder to do than

          • by zlives ( 2009072 )

            I always took an intuitive approach to physics, so a lot of current Physics/math is way to esoteric for me. Heck this octonion bit is way above my understanding... and that's when i want to understand it.
            it feels right... meh, i am not even qualified to make that statement but i always like the idea of an elegant universe.

            I do agree that yes both sides should be funded, i think our capacity to intuit is largely dependent on a basic understanding. if the basis is complex enough intuitive leaps can be made be

    • You must be joking, it only proves die-hard mathematicians have not one step progressed towards women.
  • by dtmos ( 447842 ) * on Friday July 27, 2018 @12:44PM (#57019914)

    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 [wikipedia.org],” 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.

    • I forgot about that. When I was in university and learning Maxwell's equations, it was explained to us that using Calculus was the "simpler" way - Maxwell's original method used an approach which lead to a dead end.

      Thanx for the memories.

    • by Tomahawk ( 1343 ) on Friday July 27, 2018 @01:08PM (#57020072) Homepage

      It's also interesting that the equations could be changed that way. Maybe that works between all of levels.

      Here's the Unified Theory of Everything in octonions using x formulae that explain everything.

      Now that that's done, we can simplify them into the fewer equations in quaternions.

      Now that that's done, we can simply again to fewer equations in complex numbers.

      Now we have something that's much easier to understand, but to properly appreciate it and work with it and expand upon it you need to go back to the original octonion. Then resimplify.

      (If simply is the correct word here)

      • by zlives ( 2009072 )

        all this is beyond me, however it seems that because something was harder(obtuse) it was ignored (mostly) even though it may actually had been the right path to understanding a unified theory.
        some things should be hard, which doesn't mean the application can't be simplified or made available/accessible.
        this has been a happy friday for me.

      • You could hope for that. After all, a lot of problems in signal processing are solved more easily by flipping to a different domain (say, frequency instead of time), solving it there, and then converting back.
        In fact, something similar is what Ed Witten was on about in joining the various (sub) string theories, so that a problem insoluble in one of them could be solved in another, then converted back.
      • Here's the Unified Theory of Everything in octonions using x formulae that explain everything.
        Now that that's done, we can simplify them into the fewer equations in quaternions.

        The number of quaternion equations will probably be greater than the octonion or vector versions. Which is why they were avoided for a while.

        Each equation will be quite simple, but the number will grow pretty dramatically. Compare how simple motion is taught pre and post calculus. Pre-calculus, you're taught dozens of equations

    • 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.

      With a name like that, he's got to be an Avenger, right?

    • Re: (Score:3, Interesting)

      by JaxDefore ( 5474230 )
      after years of reading slashdot I was moved to create a login because of your post. It is the single most interesting thing I have read in ages. Thank you for it - this is why I slog through the posts every day - golden nuggets of fascinating insight. it reminds me of James Burke's Connections (which I hope comes across as a positive mention to you - that's how I meant it)
    • Re: (Score:2, Informative)

      by Anonymous Coward

      According to Michael J. Crowe, "A History of Vector Analysis", Maxwell developed his theory using component analysis in the 1860's. He began studying quaternions in 1870, and presented both component and quaternionic notation in his 1873 "Treatise on Electricity and Magnetism." A brief history can be found at http://fexpr.blogspot.com/2014/03/the-great-vectors-versus-quaternions.html

      • by dtmos ( 447842 ) *

        Crowe is right; I was writing from memory and forgot the component analysis cul-de-sac -- I should have written 1873 instead of 1865 for the quaternion publication. History is always more complicated than we remember; Maxwell spent more than a decade developing the theory, and it's the quaternions that most recall of that era.

        Mea culpa.

  • by cascadingstylesheet ( 140919 ) on Friday July 27, 2018 @01:26PM (#57020156) Journal
    ... I was gonna say that.
  • by volvox_voxel ( 2752469 ) on Friday July 27, 2018 @01:31PM (#57020198)

    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... [arxiv.org]

    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/ [ucr.edu]

    "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/... [ucr.edu]

  • by thePsychologist ( 1062886 ) on Friday July 27, 2018 @01:43PM (#57020272) Journal

    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.

  • Geometric Algebra (Score:4, Insightful)

    by sfcat ( 872532 ) on Friday July 27, 2018 @01:46PM (#57020290)
    Octonions, quaternions and the like are algebras for dealing with dimensions represented by imaginary numbers. But they are special purpose, like most of the algebras used by physicists. They were 1 of 2 ways to represent vectors in math (the other being Vector Algebra). A way of uniting these two methods into 1 framework was discovered by Clifford about 50 years later but by that time there was a big split in math over VA vs Quaternions (which VA won) [youtube.com]. And most of the field ignored Clifford and his way to unite the two (VA and Quaternions). This is a big reason why its hard to unite multiple parts of physics as some still use Quaternions while most others use VA.

    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.

  • Her name sounds like a comic book super hero. Cold Fury?

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