
The Octonion Math That Could Underpin Physics - jonbaer
https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720
======
sdenton4
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
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/."

~~~
PurpleBoxDragon
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 complex number. I remember seeing some w based
number system derived from the cube root of either 1 or -1 (forget which, but
w and -w were the solutions that weren't 1 or -1), but it did all the same
things that complex numbers do and was considered mostly uninteresting.

It also seems like there is a pattern to go infinitely beyond octonions, but
they all behave identical to octonions, but are the octonions even needed in
the same way complex numbers are needed, or do they just make some math
problems easier to work with?

~~~
yaks_hairbrush
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 couldn't do it. But he did find 4-dimensional
numbers -- the quaternions. And they were really neat. You can use them quite
well for classical mechanics, and electromagnetism, and even special
relativity. So, why don't we?

We look at Maxwell's equations of electromagnetism today, and they're really
nice, single-line vector formulas. You can also write them as nice, single-
line quaternion formulas. Our notion of vector didn't exist at the time the
quaternions were first used, and it was a boon to have quaternion notation to
simplify some of these physical laws. Vectors and quaternions competed for a
bit, and vectors won since they generalize to arbitrary dimensions.

Hidden inside of quaternion multiplication, you can find the three-dimensional
versions of the dot product and cross product. And they do have some
theoretically interesting properties for number theory and abstract algebra.
In the end, however, sometimes items are discarded in favor of better items.
I'd rate quaternions as one of the coolest items that ultimately wound up in
the discard pile.

~~~
XorNot
Quaternions show up all the time in 3d graphics - they're how you represent
rotation matrices without the problem of gimbal lock.

I don't know if that's significant but it was how I stumbled on the concept in
high school when I was messing with Direct X.

~~~
kbwt
Rotation matrices by themselves do not suffer from gimbal lock. I think you
meant Euler angle representations with the rotations always applied in a
consistent order around the pitch/yaw/roll axes.

~~~
yoklov
Right. Just to add on, one reason Quaternions are still used in graphics
(despite rotation matrices not suffering from gimbal lock either) is that
they're easy to interpolate between, even if you have many.

If you just have two you can slerp (or not), but if you have a large number of
them (weights from an animation system, for example), a basic weighted sum
followed by normalizing is shockingly well behaved and extremely fast.

~~~
andybak
This is a timely comment. I've been aware of quaternions for rotation for
decades but only learned yesterday that rotation matrices can be used for some
use cases. I'd never heard this small fact until then and now I stumble across
your comment!

------
anaphylactic
> 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. With octonions you've already lost associativity and
commutativity, though.

~~~
Maybestring
>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

~~~
OscarCunningham
Another (related) property that fails is that inverses stop being useful for
cancellation. Inverses still exist, for every p there's a q with pq = qp = 1,
but if you've got an equation ap = b you can't cancel to get a = bq, because
we don't have associativity. The left hand side (ap)q doesn't equal a(pq), so
you can't reduce it to a.

Of course associativity doesn't hold in the octonions either, but it holds
just enough for cancellation to work.

~~~
anaphylactic
Yeah, octonion multiplication is alternative.

------
ianai
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.”

~~~
_bxg1
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.

~~~
madhadron
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.

~~~
ianai
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.

~~~
btrettel
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.

~~~
ianai
Either applies really.

------
chunky1994
It's rather amusing that the author assumes that non-associtiave objects are
"weird" for physicists (or at least that was my reading), since the velocity
addition formula is in general non-associative and that has been extensively
studied.

(I remember three separate occasions in my undergrad particle physics class
where we actually went through all the calculations involved with the velocity
addition formula and finally saw that SOL was wacky).

[https://en.wikipedia.org/wiki/Velocity-
addition_formula](https://en.wikipedia.org/wiki/Velocity-addition_formula)

I wonder if that is uncommon or if the article is just a bit ungenerous
towards our understanding of non-associative objects.

~~~
ssivark
To be fair, one never talks of relativistic "velocities" beyond undergrad
physics. It is much easier to talk about Lorentz "boosts", and as members of a
Lie Group, they are associative, though not commutative.

~~~
throwaway37585
This. It's better to just make the full leap into spacetime geometry
(4-vectors and 4-tensors) than keep trying to slice it into annoying
3-vectors.

------
adamnemecek
I’ve been exploring this idea as well however I have a hunch it’s not actually
octonions but dual quaternions as they are the perfect formalism for
representing 3d movement over time.

And to add to that, the are a Lie group I.e. they are anticommutative I.e.
AB=-BA.

I’ve also been exploring this relationship between dual quaternions and linear
logic. It’s pretty wild. I’m curious if anyone has any opinions on this.

~~~
stochastic_monk
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.

~~~
adamnemecek
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/](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.

~~~
electricslpnsld
> [http://www.chinedufn.com/dual-quaternion-shader-
> explained/](http://www.chinedufn.com/dual-quaternion-shader-explained/)

Offset curve deformations blow dual quaternion's out of the water for rigging!

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

~~~
electricslpnsld
Fast enough for use in live character rigs in film running on the CPU -- I
know the technique is used at Sony Imageworks and at Dreamworks. I can't
comment on whether the method is suitable for live rigs in games, or how it
would translate the GPU.

------
steamer25
This reminds me a bit of atomic orbitals and how they're based on spherical
harmonics. I was curious why there were e.g., eight electron 'sockets' in the
second shell. Eight seemed like a very arbitrary number to me and my high
school chemistry teacher's inability to explain at the time did it's share to
put me off chemistry.

Many years later I remembered my old question and started looking it up. It
turns out that eight is the sum of 1+3+3+1 perhaps similarly to what's in the
article.

Spherical harmonics ends up giving rise to a three dimensional 'overtone'
series (borrowing from my understanding of music theory). In the first order
there's only one mode of vibration. In the second order there are three
additional modes. The summands above are something like positive and negative
degrees of freedom for each mode in the second shell.

Here's a diagram of what the modes look like in each order:

[https://i.ytimg.com/vi/OkDYbIhisZE/maxresdefault.jpg](https://i.ytimg.com/vi/OkDYbIhisZE/maxresdefault.jpg)

...and here's an animation of a sphere undergoing the differing modes of
vibration:

[https://youtu.be/EcKgJhFdtEY](https://youtu.be/EcKgJhFdtEY)

If I understand correctly, the math related to atomic orbitals can be
described with 3 dimensions of space: x, y and z plus one more orthogonal
dimension of frequency/time which would mean quaternions would be most
directly applicable?

------
ChrisLomont
"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. "

Not true. Any field (in the algebra sense) has these properties. [1]

Quaternions and octnonions also have weirder properties: quaternions are non-
commutatve (j _k=-k_ j) and octnonions are non-associative: a(bc) != (ab)c.

I think the article meant these are the only Euclidean Hurwitz algebras [2],
which is a far cry from the claim.

[1]
[https://en.wikipedia.org/wiki/Field_(mathematics)](https://en.wikipedia.org/wiki/Field_\(mathematics\))
[2]
[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...](https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_\(composition_algebras\))

~~~
pishpash
It's in the article towards the end, but these are the only kinds of ... "over
real numbers," which constraint Furey believes may be only an approximation.

~~~
HerrMonnezza
Even when adding "over the real number", the claim in the article is wrong:
R(x), the algebra of rational functions of one variable with real coefficients
is a field, and a module over R containing a isomorphic image of R as a
subfield.

You really do need the additional constraint of the Hurwitz form to restrict
the possibilities to R, C, H, O ...

(Of course this has nothing to do with the work of Furey - I'm just seconding
that the claim _in the article_ is incomplete and inexact as worded.)

------
scotty79
> “Because while it’s very easy to imagine noncommutative situations — putting
> on shoes then socks is different from socks then shoes — it’s very difficult
> to think of a nonassociative situation.” If, instead of putting on socks
> then shoes, you first put your socks into your shoes, technically you should
> still then be able to put your feet into both and get the same result. “The
> parentheses feel artificial.”

> The octonions’ seemingly unphysical nonassociativity has crippled many
> physicists’ efforts to exploit them.

I'd say nonassociativity is extremely physical property. And example with
shoes and socks, if anything, shows just that. In idealised mathematical
model, if you put socks inside shoes and then insert feet into socks you could
end up with same thing as if you put feet in socks an then both in shoes. But
in real world or just accurate model of physical world you'll end up with very
different result.

I feel that nonassociativity is what's missing to take into account time which
unidirrctuonality is suspiciously missing from almost all of the physics.

~~~
politician
If you're looking for a deep look at time -- a Total Perspective Vortex, I'd
recommend "Spontaneous Inflation and the Origin of the Arrow of Time".

[https://arxiv.org/abs/hep-th/0410270](https://arxiv.org/abs/hep-th/0410270)

~~~
scotty79
I don't know. Unidirectionality of time just feels like something that should
be everywhere. Nonassociativity of basic math that governs all of matter seems
for me to be a better source for it than inflation or entropy or gravity or
any one specific thing.

------
imglorp
How does this tie into E8? Does this work validate Garret Lissi or is it
parallel?

[https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory...](https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything)

~~~
atonalfreerider
I remember hearing Garret Lisi's TED talk on the E8 ToE (link below) years ago
and hoping that components of the theory would eventually come to light
through advances in particle physics research. Has anyone heard anything
supportive or to the contrary since?

TED Talk:
[https://www.ted.com/talks/garrett_lisi_on_his_theory_of_ever...](https://www.ted.com/talks/garrett_lisi_on_his_theory_of_everything)

Paper: [https://arxiv.org/abs/0711.0770](https://arxiv.org/abs/0711.0770)

 _Edit_ I remember seeing an update from him two years ago where he addresses
some of the criticism: [https://www.quora.com/What-is-the-status-of-Garrett-
Lisis-E8...](https://www.quora.com/What-is-the-status-of-Garrett-
Lisis-E8-Theory)

~~~
ansible
There's been some criticism of his work, back when it first came out.

I'm fuzzy on the details now, but the criticism was along the lines of "mixing
things together that don't make sense".

~~~
Cobord
I remember something about being unable to do any CP violation.

------
amai
Why are complex numbers not called duonions then?

~~~
OscarCunningham
And why are the real numbers not called "onions"?

~~~
kibwen
Obviously due to potential confusion with the zero-dimensional numbering
system, "0nions".

~~~
dahart
Hehe, well we could have unions & oneions. For some reason as I'm reading
about octonions, I keep picturing 8-cloved garlic.

------
whatshisface
I think that maybe if you lock a mathematical physicist in a box forever with
an x-TeV collider without letting them upgrade it, they will eventually find a
theory that hits all of the datapoints and depending on their philosophical
weakness then declare it "final."

~~~
OscarCunningham
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.

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

~~~
tlb
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.

~~~
whatshisface
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.

~~~
RobertoG
I think you are wrong. If he is following the scientific method, and he
doesn't have evidence, he should 'conclude' that.

The key is that in science, you never 'conclude' (in the sense of finalize)
anything. Everything is temporal until new evidence deny your current
understanding.

------
Ono-Sendai
I have to disagree that quaternions underlie Special Relativity. Although
special relativity does use 4-vectors, those aren't quaternions.

~~~
madhadron
I remember going down that path when I was an undergraduate. Fortunately I had
a very experienced theorist to hand who explained that, yes, people did try
that, but stopped bothering because it doesn't generalize to general
relativity, and there was no point keeping two mathematical toolboxes around
when you could have one.

~~~
eigenspace
Clifford algebras are great for both SR and GR and have all the properties
that one likes in quaternions (and more)

------
mikorym
As a math graduate, octonions (and quaternions) at least to our group, was
something that got discussed once or twice and not much after that. However, I
would not call them weird or unusual, at least not more _weird_ than something
like a near-ring which similarly simply drops one of the common assumptions
for rings (I'm not sure which, actually). As mentioned in some of the
comments, nonassociative fields do get studied. Studying not necessarily
commutative structures such as (all) groups is in fact an even much more
common thing to do.

I would say that studying octonions exclusively would be something I
personally would avoid, as I would rather try to study the four structures
(reals to octnonions) together, either more generally (e.g., group theory and
ring theory) or more abstractly (e.g., as members of categories) and form an
opinion on whether I think octonions in particular are useful for the
questions that I want to ask.

That is not to say that the research here is not interesting, but it is
difficult to judge that from "popular" mathematics articles. I got the
impression that the author of the article places a much higher priority on the
pictures accompanying the post.

I remember that I found it interesting that studying the four dimensional
spacetime bears more fruit than stopping at three space dimensions, and that
at the same time from complex numbers the next structure ends up also having
four dimensions (i.e., being modelled by 4-tuples). However, apart from being
interesting in this narrow sense, I do not know whether this suggests any
creative yet precise mathematical questions.

------
fouc
Why do I feel like the octonions are likely related to the spin networks of
LQG (loop quantum gravity). The non-associative property is probably key to
explaining how change in space is propagated through the spin networks. And
thus why time is directional..

------
danbruc
In the approach from the article it seems as you are picking a mathematical
structure in the middle of nowhere with the universe faintly visible at the
horizon and then you start wandering around hoping that you will stumble
across a path leading to the horizon.

But there don't seem to be many reason to believe that such a path exists,
there are countless mathematical objects you could pick as a starting point
and almost none of them have a path leading to the universe at the horizon.
And even even of those with path going roughly into the right direction many
will take a turn before you arrive at your destination.

So it seems much more promising to me to start at the universe and the
mathematical structures describing our observations in a straight forward way
and then explore from there the surrounding mathematical structures to see if
they are a better fit, suggest new ideas, or whatever.

~~~
sannee
> In the approach from the article it seems as you are picking a mathematical
> structure in the middle of nowhere

Quaternions were very popular way of expressing the "classical" physics around
the 19th century (and the vector algebra we know today is in some ways just a
derivative of quaternion algebra). Complex numbers are extremely useful in
many fields even today. It's hardly in the middle of nowhere.

~~~
danbruc
But the article is not really talking about quaternions - which are surly a
useful tool, probably best known for the nice way in which they can describe
rotations - but about R⊗C⊗H⊗O. And it's at the very least not obvious that
this thing is anywhere close to where they journey is hopped to lead to.

~~~
sannee
Octonions are to quaternions as are quaternions to complex numbers (and
complex numbers to reals). This is called "Cayley construction" iirc.

~~~
danbruc
Sure, but the structure the article is about is the tensor product of the four
algebras which is an algebra with 64 real dimensions.

------
stultifying
The really interesting part in all this, is that we're still reduced to
bouncing values off of imaginary numbers to obtain accurate readings.

We toss our known quantities into a void, anticipating that if some
impossible, imaginary thing really can fill the gap, and if or when it does,
we'll catch the rebound off of it, and the rest of the universe proceeds
predictably.

Somehow, we're always put into a position where we have to close our eyes, fly
blind for some undisclosed intervening moment of unspecified length, and when
we open our eyes again, we're grounded by familiar territory again.

It really is kind of stultifying.

------
pooya13
I never liked the name complex numbers. Something like dual numbers would have
been much nicer and more descriptive.

By the way any idea why they don't mention the sedenion numbers?
[https://en.m.wikipedia.org/wiki/Sedenion](https://en.m.wikipedia.org/wiki/Sedenion)

------
kentbrew
Pachimari, of course, is the official mascot of octonions everywhere:
[https://kotaku.com/overwatchs-little-onion-octopus-has-
becom...](https://kotaku.com/overwatchs-little-onion-octopus-has-become-a-
sensation-1820855871)

------
marlag
> They “imagined that the next bit of progress will come from some new pieces
> being dropped onto the table, [rather than] from thinking harder about the
> pieces we already have,”

When the best minds we have divide into two camps, one saying that (a) "with X
and Y we don't have enough information to solve Z" and the other saying (b) we
do but we need to think harder, the first camp builds a particle collider, the
other creates what, string theory? Aren't we doing "fuzzy science" here?

It seems that the experts in one of the camps should go back and retrace their
steps because somewhere along the line they made an assumption based on some
data that they (I assume) forgot to encode into their equations and now they
have trouble taking it to the next level. Why is it not clear to us that eiher
a or b is true?

------
pishpash
So, did Furey get a faculty job or can she be viewed with her accordion
somewhere?

~~~
selimthegrim
FWIW, I'll certainly tell my department chair in New Orleans about her if
they're doing another faculty search soon. Maybe she can have her cake and eat
it too that way.

------
stcredzero
I wonder how many hardcore Harry Potter fans are smiling at the 1st
photograph?

------
starchild_3001
This article reminded me of wave function being a complex-valued function!
Does this have anything to do with octonion, nature's symmetries/invariants or
the standard model?

------
didibus
Are we getting to the point where physics actually isn't showing any more
useful properties for us to leverage?

Isn't that kind of what it would mean if it turned out octonions were at its
core?

------
fifnir
A layman's quest to understand wtf this is...

>In mathematics, the octonions are a normed division algebra over the real
numbers .

wtf is a normed division algebra??

>In mathematics, Hurwitz's theorem is a theorem [...] solving the Hurwitz
problem for finite-dimensional unital real non-associative algebras endowed
with a positive-definite quadratic form.

... right.

I have the same problem when I try to understand anything statistics related,
I get hit by a barrage of unknown words and my brain just melts.

Is there any place that explains mathematical concepts in ... different ways?

~~~
Micoloth
Well yeah, math books...

Jokes aside, i too think we need a new framework for divulgating math that
actually tells what you need to know, without just handwaving at it, BUT
without the amount of technical details of a mathematics class.

What do you know, i think this is Possible, too.. You can communicate a
surprising amount of information if you use words properly.

Of course, since this has never been done except from basic maths, it would be
quite a task to embark in, and one would only do it if it made economic
sense..

~~~
sigstoat
> Jokes aside, i too think we need a new framework for divulgating math that
> actually tells what you need to know, without just handwaving at it, BUT
> without the amount of technical details of a mathematics class.

oh man, have i got bad news for you about the amount of technical detail
present in math classes.

------
bvinc
Can anyone expand on the current successful similarities between octions and
the standard model in plain English?

------
lifeformed
If it goes: 1D, 2D, 4D, 8D, why does it stop at 8? Is there a hexadeconion?

~~~
danbruc
They are called sedenions [1] and you can repeat the construction taking you
one level up [2] infinitely.

[1]
[https://en.wikipedia.org/wiki/Sedenion](https://en.wikipedia.org/wiki/Sedenion)

[2]
[https://en.wikipedia.org/wiki/Cayley–Dickson_construction](https://en.wikipedia.org/wiki/Cayley–Dickson_construction)

------
dbdoug
'Looking like an interplanetary traveler, with choppy silver bangs that taper
to a point between piercing blue eyes' WTF? for a few minutes, I thought this
was a serious article by a serious journalist. Apparently not.

------
forkandwait
If anyone can make octonions cool it would that physicist

------
c54
Her name is Cohl Furey, that is SO COOL!

------
csoroz
[https://motls.blogspot.com/2018/07/cohl-furey-understands-
ne...](https://motls.blogspot.com/2018/07/cohl-furey-understands-neither-
field.html)

