The Octonions (Real Numbers Are Trivial)
July 22, 2018 8:11 AM   Subscribe

The Peculiar Math That Could Underlie the Laws of Nature - "As numbers go, the familiar real numbers — those found on the number line, like 1, π and -83.777 — just get things started. Real numbers can be paired up in a particular way to form 'complex numbers', first studied in 16th-century Italy, that behave like coordinates on a 2-D plane. Adding, subtracting, multiplying and dividing is like translating and rotating positions around the plane. Complex numbers, suitably paired, form 4-D 'quaternions', discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiseled the formula into Dublin's Broome Bridge. John Graves, a lawyer friend of Hamilton's, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space." (via)
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. 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?
also btw...
-Cohl Furey: Division algebras and particle physics
-Split Octonions and the Rolling Ball
-John Baez on the number 8
-octonionic probability
-p-adics
posted by kliuless (26 comments total) 60 users marked this as a favorite
 
This is an imposing topic to chime in on and I know almost nothing about the physical models under discussion and only the rudiments of the math, but as a lapsed mathematician I love this. The quote from the first link, "the model that, in hindsight, feels inevitable," is so accurate as an encapsulation of how beautiful math can be when it somehow maps perfectly to observable phenomena.

This seems a little like what I understand string theory (which gets a passing mention in that first link) to have been: a model that was cooked up because it was elegant and beautiful but didn't pan out so much on the accurate side of things. And reality does seem stubbornly resistant at times to falling in line with the perfect mathematical models in our heads, so in my ignorance and skepticism I don't have super high hopes for this either. But there are enough fundamental-seeming, curious and beautiful mathematical objects here that I really want to believe it will produce real results!
posted by valrus at 8:57 AM on July 22, 2018 [2 favorites]


Where is Dr. Otis Eugene "Gene" Ray when we need... oh sigh...
posted by sammyo at 9:01 AM on July 22, 2018 [2 favorites]


To me, quaternions were those completely incomprehensible, unusable alternatives to a 3x3 or 4x4 matrix in 3D graphics.
posted by Foosnark at 9:03 AM on July 22, 2018 [3 favorites]


Oh and gimbal lock.
posted by sammyo at 9:03 AM on July 22, 2018 [2 favorites]


kalessin: It was always entertaining in grad school to find out almost every branch of science was using the same math, just evaluating results in different contexts. I worked in dynamics & controls, which basically use the spare dimensions to evaluate stability or transform problems that appear underrepresented by data into ones that can be understood and evaluated in some other context.

I have fond memories of discussing research with a math post doc (good to have a regular running partner) and getting major insights into results by understanding the math better. He really had to dumb his work down for me to understand any of it, but I like to think that helped him when he published.
posted by ptfe at 9:53 AM on July 22, 2018 [4 favorites]


how meta is it that her first name looks like "complex quarternions" in her blackboard notation
posted by eustatic at 9:54 AM on July 22, 2018 [2 favorites]


Thomas Pynchon explores quaternions in Against the Day
posted by chavenet at 9:59 AM on July 22, 2018 [2 favorites]


It was always entertaining in grad school to find out almost every branch of science was using the same math, just evaluating results in different contexts.

yeah, and it's physics math. applying physics math to say, psychology has always seemed like a stopgap solution at best.
posted by eustatic at 10:30 AM on July 22, 2018


I love how we drop fundamental properties of numbers as we go up/down in the series. We lose the triangle inequality when we go from the Reals to Complex numbers. We lose Commutivity when we go from Complex numbers to Quaternions (a*b does not have to equal b*a) and Associativity with the last jump from Quaternions to these kind of numbers. That's always felt deeply meaningful to me.
posted by aleph at 10:30 AM on July 22, 2018 [10 favorites]


We lose Commutivity when we go from Complex numbers to Quaternions (a*b does not have to equal b*a) and Associativity with the last jump from Quaternions to these kind of numbers. That's always felt deeply meaningful to me.

We bid a fond adieu to Groups, then?
posted by jamjam at 11:03 AM on July 22, 2018 [3 favorites]


The algebras of normed division and Clifford part ways.
posted by LarsC at 11:14 AM on July 22, 2018 [1 favorite]


The wonderful thing is, that if you dig hard enough at the roots of mathematics, there are always layers underneath where discoveries can be made, instead of finding out that Carl Gauss knew what you just figured out when he was four years old.
posted by delfin at 11:19 AM on July 22, 2018 [2 favorites]


John Baez has done a lot of work (that I'm trying to understand in my own amateurish way) on octonions and relating division algebras to quantum theory and supersymmetry.
posted by LarsC at 11:23 AM on July 22, 2018 [2 favorites]


Hm, we use dual-quaternions to represent position and orientation for the calibration slerp on our 6DOF robots. Those have apparently been called octonions in the past (says wikipedia), but modern octonions are something different again.
posted by rhamphorhynchus at 11:28 AM on July 22, 2018 [2 favorites]


Off topic: I just found an interresting tid-bit. John Baez is a cousin of Joan Baez the singer.
posted by Increase at 12:30 PM on July 22, 2018 [4 favorites]


I thought this topic was about eight onions...
posted by Quackles at 12:52 PM on July 22, 2018 [3 favorites]


I can't even get my head around quaternions, so I don't think I have any chance with octonions.
posted by runcibleshaw at 1:13 PM on July 22, 2018 [1 favorite]


runcibleshaw: Yes. a*(b*c) maybe not equal to (a*b)*c ...[shakes head]
posted by aleph at 4:06 PM on July 22, 2018 [2 favorites]


Not being able to follow the maths, I was instead impressed by seemingly uncorrelated things, such as the significance of the number 8. This reminded me that all I ever got out of trying to understand Lisi's proposal was that it somehow had an 8-ness, which sounded like the Eightfold Way and that was good, so maybe these other things with lots of eights would also be good.

The other thing that seemed related to this was that Cohl Furey sounded like what would happen before Cold Fusion, so probably that counts against the eight-ness and means the likely outcome here is that in about eight years there will be an article like this one using a variant on the cold fusion pun in the headline.
posted by pulposus at 4:49 PM on July 22, 2018 [1 favorite]


I'm a natural born cynic so the idea that the universe is describable and quantifiable, yet down the centuries we're using the best available but "wrong" type of numbers appeals entirely too much. Anything this amazingly mathy also makes me regret majoring in Metallica over Maths. Can't be using them old apple-counting, sorrow-harvesting numbers to describe the universe, tsh!

I'm off to divide twenty-two by seven, cross my fingers hard and maybe read a few paragraphs into some of these links.
posted by I'm always feeling, Blue at 5:39 PM on July 22, 2018 [1 favorite]


Off topic: I just found an interresting tid-bit. John Baez is a cousin of Joan Baez the singer.

I'll be damned, here comes your spectral sparsifier again...
posted by otherchaz at 7:20 PM on July 22, 2018 [3 favorites]


Sometime in the last few months, I came across a beautiful explanation of how quaternions arise as rotations in 3-space and how the 4th coordinate is redundant but makes the math more friendly. Unfortunately, I've completely forgotten where I saw this, so I can't share it. :-/

The TL;DR is that a rotation in 3-space is defined by a 3-vector (the axis of rotation) plus the (scalar) angle of rotation. But you don't actually need to encode the angle of rotation separately from the 3-vector: the magnitude of the 3-vector doesn't represent anything, so we can repurpose it to encode the angle of rotation. This leads to a very natural mapping between 3-vectors and quaternions with real part 0, because the (i, j, k) basis vectors of 3-space (the axes for rotations in the yz, xz, and xy planes) are directly analogous to the (i, j, k) orthogonal roots of -1 in quaternions.
posted by chronostachyon at 8:18 PM on July 22, 2018 [2 favorites]


Yay, something to watch later. I only remember quaternions from a long time ago (3d Graphics Gems era) so I don't know If I'll be lost or it will make sense.

But for I'm always feeling, Blue's 22/7reference... Infinite fractions and the most irrational number... know the why of 22/7.
posted by zengargoyle at 8:26 PM on July 22, 2018


A while ago I wrote a (hopefully) intuitive explanation of how quaternions represent rotations in three dimensions, along the same lines as chronostachyon's comment. Maybe someone here finds it interesting?

That said, I have no understanding of the octonions. But if John Baez likes them then they must be cool.
posted by a car full of lions at 4:26 AM on July 23, 2018 [1 favorite]


(though on further reflection, chronostachyon's talking about the purely imaginary quaternions, whereas my link is about the unit quaternions -- no big deal, because the former is the Lie algebra of the latter, and I'll shut up now before I entirely alienate everyone here)
posted by a car full of lions at 4:31 AM on July 23, 2018 [2 favorites]


"Octonions" sounds like a rejected Terry Pratchett idea.
posted by Chrysostom at 3:40 PM on July 24, 2018 [1 favorite]


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