Quaternions, THE Numbers of the Universe
By Doug Sweetser | March 26th 2012 11:54 AM | 42 comments | Print | E-mail | Track Comments

Trying to be a semi-pro amateur physicist (yes I accept special relativity is right!). I _had_ my own effort to unify gravity with other forces in...

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The title of this blog is the title of a talk I will be giving real soon, this Thursday, March 29, as part of "Ignite Boston 9" at MIT's Media Lab.  No, you cannot go unless you are one of the 400 people who signed up, but video should be available soon enough.  Ignite talks short, all of 20 slides, 15 seconds a slide, so 5 minutes total.  Since the slides auto-advance, the time limit is strictly enforced.  The topics of discussion are all over the nerd map.  I am pretty sure I will be the only one to discuss number theory in any way.

[Update: as a commenter pointed out, here is the actual talk.

The title of each slide will be written in bold.  I read a book on presentations that recommended most slide titles should be sentences, so this transcription of a future talk is easy, no wormholes needed.

Quaternions, THE Numbers of the Universe. The title may be true someday. I am leading a nascent open science project to see if the title is true.

Quaternions are numbers with four parts.

The numbers can bee added, subtracted, multiplied, or divided. That is both ordinary and a bit odd. Let's find a real-world example of this kind of time and space addition.

We are here now.  Add 6 hours and various paths, and all here now will be asleep.  Both time and space are added together to make a statement about a future place.

Blame the Greeks with their search for permanent perfection

This is pre-Euro zone, more like 300 BC. Axioms, postulate, and proofs were both a great advance and a tight straight jacket.

Euclidean lines are straight yet Nature is full of odd lines.

Fractals are the math of rough. It was a tough project to sell the idea of fractals to professional mathematicians, even though Benoit Mandelbrot was every bit the pro.

Geometry is permanent, yet everything is temporary.

We need the math of transience. Time should not be treated as a tacked on parameter. Instead it should be in the center of the algebraic stage.  I hope quaternions are up to the task.

Quaternions have a horrible rep that continues to this day.
"Quaternions...though beautifully ingenious, have been an unmixed evil to anyone who have touched them in any way, including Maxwell."
--Lord Kelvin
Some people today would include me on such a list.  I prefer my evil unmixed.  Nothing like being bitch slapped from the past.

Quaternions are a one-trick pony for doing 3D rotations.

Great for games and rocket science, but they cannot do a Lorenz boost* which is a technically solid reason enough to never use them for a serious calculation in physics.

Quaternions are magic math.  Square one and sees the square of the invariant interval of special relativity.

Take a simple derivative and see the fields of EM:

Analytic geometry is math making a graph.  This is a real simple graph, the venerable straight line. It is a static graph that is exclusively about space.

Analytic animations is math making a movie.

A dull but dynamic image about spacetime.

Time reversal requires one to remember.

Time sits in the real number position of a quaternion.  Real numbers are time in the analytical animation representation.

Space reversal requires a mirror reflection.

Imaginary numbers are about 3D space.

The difference between real and imaginary numbers makes sense.  Compare the two animations above.  No more need to reference the square root of -1.*  Using one's memory for real number reversal versus a mirror just looks different from the mirrors required for imaginary number sign flips.

* The square root of -1 truism is still true.  The Argand plane is still true.  I found the 90 degree difference between the real and imaginary axis to be both useful and hollow at the same time.  For me, the animation representation of the same algebra makes more sense and feels solid.

Time AND space reversal can look like the future.

Think about the red line being both a reversal in time and in space of the yellow line while smoking some weed.  Please do so under medical supervision.

Most of my time is spent on a new field theory for gravity.  I call it a failure in progress :-)  I have been able to derive the Maxwell equations using quaternions, something Maxwell himself wanted to do.  I still have hope for finding a legal variation on Max.

Blog Monday night on Science20.com.  And it usually creeps into Tuesday morning.

The blogs get a thousand reads a week, a factor of 20-100x of the lead bloggers on the site.  The work has worth it since it provides a good source for critical feedback.

A small group is working on this topic without funding.  They hail from the US, UK, and Australia.  The sparse group was recruited from investments I have made on the web.  My YouTube videos have generated 235k downloads.  Quaternions.com is on the first page of lists for the word for four Roman soldiers as found in the King James Bible.  VisualPhysics.org has all the Drupal tools for a community, just not the people.

Quaternion jam sessions are on Saturdays at 11am Eastern Standard Time. It is a Google+ Hangout.

I hope to see some of you there.

And that should do it for the 5 minute talk... It will be interesting to see if I have made it too technical or not, how much info can be shared with such a crowd.

Doug

Snarky puzzle: Is an animated circle symmetric in time? Is it symmetric in space?

Next Monday/Tuesday: The little gauge that could

Talk Report:

The Ignite Boston 9 event was a good way to spend an hour and a half, so long as you don't mind wild shifts in subject.  Some have complained about the shifts within my own talk, but in a way that fit with the theme of the evening, which there wasn't one except these were subjects the presenter cared about.  The first talk was about history, who owns it.  There was a talk about making buildings adjustable for different uses.  A cute robot captivated people for five minutes.  One person focused on revolution.  Naturally there was a talk about using GPS units to map towns in Uganda.

Five or six people gave me positive feedback.  That must be taken with a block of salt as negative feedback after a talk is rare unless it is on a cultural flashpoint, which recruiting people to work on quaternions is not.

This talk represented marketing work.  I passed out all of 1 business card.  I am not a good pitchman since I point out flaws.  I still consider it a success because one cannot know what secondary or tertiary conversations might happen because I was one of the dozen or so people who stood up in front and talked for five minutes instead of the two hundred (estimated) in the crowd.  The talk was recorded, so will have a web life that is not predictable, but probably will not amount to much.

The moderator kept pitching us to eat the pizza, but there was not much mingling (not that I am good at that either).  I did have more conversations about the subject than happens at an early morning APS meeting.

I am keeping the day job, but it was worth the effort.

I need to get a quick rand out of the way:
A quaternion is a number (a scalar), and you destroy their mathematical uniqueness/usefulness as soon as you try to promote them to four-vectors, parts of vectors, parts of tensors, etc.

That is a long discussion, so moving onto something that is more immediately useful, let me try to give some honest neutral advice on your presentation choices.

"It will be interesting to see if I have made it too technical or not"

As is, you aren't even saying anything. If I pretend I hadn't read your stuff here so I couldn't guess what you were trying to convey with each snippet, the only information you successfully conveyed to me in that talk was to advertise your presence on the internet. It works as an ad, but that's about it.

Definitely get rid of those animations. I know you love them for some reason, but even after you were asked for clarification here where people have time to discuss, it is not clear at all what information you are trying to convey with them. It would be like if I wanted to talk about complex numbers, noted they have two components, and then kept showing the audience various 2-d parametric plots (x(theta),y(theta)) animated by drawing them from theta=0 to 2 pi, and claiming each graph somehow gave insight into complex numbers. It is nonsense. It's a way of drawing a parametric plot. It contains no extra information than the parametric plot itself. Furthermore the parametric plots you chose are completely uninteresting. I fail to see why you continue to feel such "deep meaning" from your animations. There is absolutely no connection to quaternions beyond the trivial : a quaternion has four components, and the parametric plot is in four dimensions. (That random flickering "one state" part of the animation you always show is also distracting and makes it even less clear what information you are trying to convey with these.) I seriously suggest dropping the animations.

I think you have a lot you want to say, and you are trying to summarize all of it. Imagine summarizing each chapter in a book with just two words and then combining this to summarize the book ... it would be pointless in conveying information.

Even the simple concept of what a quaternion even is, such as: it has four components (you did say this), and you want to treat those as describing time and 3 spatial dimensions ... is not immediately clear in your talk. I know what you mean, but do you think an introduction with 15sec of your google map and a joke example of people's paths to their location in 6 hours is really the best way to introduce this?

Why mention the greeks? What does that add?
The fractals thing seems completely non-sequitor.
The geometry comment comes out of no-where and while we here know what you are hinting at, it is more like an inside reference for those that know than it is conveying information.
The rotations would be a good thing to explain one case where they are relevant/used today -- give this its own slide instead of killing the punch by using it as a throw away fact to lead into spacetime rotations (you don't call them spacetime rotations, so no one will even get why that leap is anything but a non-sequitor).
If you really want, then have a slide saying you extended this to spacetime rotations which is the math of relativity (I of course object to this treatment, but I'm trying to advise from a neutral standpoint here).
Then maybe a slide saying considering a field of these at every point in space-time -- like the classical electromagnetic potential -- you were also able to use them to get all of classical electromagnetism (again I have serious complaints about this, but it's your talk).
The explanation of geometry is watered down to the point of being useless there, drop that slide.
Why waste precious time showing a snippet of history of quaternions with no context to judge it, and even worse it is dissuading interest in what you are promoting (why do "fringe" physicists feel the need to "brag" about how they were already dismissed a century ago?) -- if you are going to sell a product, sell it.
The animations convey no useful information, as mentioned above, I'd really just get rid of those.
Why the hell are you talking about your music in the middle of a quaternion intro that you were already given incredibly short time to present?

At most reduce your ads to maybe a single slide at the end saying your lofty goal is to extend this to describing gravity and where they can read more. A teaser, the end.

You have very little time, so you need to be focused. Choose the one point you want to convey. Teach that one point. You are all over the place and instead of conveying more, end up just conveying less.

The site marked this as spam, how ironic.  "neutral advice" - you are joking? No, you are serious.  I will deal with one issue.
Yes, those are parametric plots based on one variable. Consider just one generalized point of the animation:

(a, b)

Time reversal will generate the point:

(-a, b)

while space reversal generates:

(a, -b)

So the time reversal plot will have:

(a, b) and (-a, b)

while the space reversal has:

(a, b) and (a, -b)

Is there a visual difference between the two?  In the complex plane which are in my 6 panel animations, the two are 90 degree rotations of each other.  That kind of representation is still in use today.  With the animation, the time reversal requires storage of information about one event to plot the other at a later time. The space reversal requires 2 events on the screen at the same time.  Real number reflection looks different from imaginary reflection in a parameterized quaternion animation.

The spirit of Lord Kelvin lives on.
PT symmetry but no C symmetry. I'm thinking that quaternions have a handnesses built in. The directions that make up i,j,k, could orientate in a left or right handed direction, a space conjugates reverses the handiness, but quaternions maths uses conjugations all over the place, with out clearly seeing weather the scalar or vector was psuedo (left right invarient) or plain (left right varient) and this is very important in physics.

If done carefully and consistently, the results will fit. Not only conjugations play a role. Also three independent reflections play a role. Each of them switches handedness. Together they constitute eight different sign choices. Not all correspond to a switch of handedness.
The Dirac and Majorana equations each couple two quaternionic probability amplitude distributions. With eight possible sign flavors for each quaternionic probability amplitude distribution this can be extended to sixty four different cases. This is enough to cover a large set of different particle types.

If you think, think twice
" "neutral advice" - you are joking?"

I made it clear my opinion of your ideas, and that was not neutral. But I said I'd give neutral advice on your presentation, and I did. I gave an assessment of your presentation from the neutral stance of: regardless of the correctness of the material, does the presentation clearly convey the information about quaternions and why they are "THE numbers of the universe".

You have only 5 minutes yet you waste many slides on things having nothing to do with your topic. Telling you to focus and remove those diversions is a very realistic statement and a completely neutral on the content of your opinions. Heck I even suggesting adding some stuff that would help your presentation, yet that I object to the content of.

How does bringing up fractals, or mentioning you play music, have ANYTHING to do with quaternions? Even if you feel it is important, at least understand that WHAT information you feel connects these topics is not clearly expressed. So regardless of what you meant to do with those slides, it it failing.

"Yes, those are parametric plots based on one variable."

And they tell the audience nothing about quaternions. I could also draw a parametric plot that is a spirograph that looks like a five petaled flower, and comment that it has the symmetry of the D_5h point group. Does that show anything useful for your presentation on quaternions? No. You waste an inordinate amount of time on these animations because you think they are conveying some deep meaning. They are not.

You should be asking yourself: What information do I want to convey to the audience?
You are all over the place, so I'm not sure you even have a clear answer to this for yourself.
Once you have and answer, then look at your slides. Assuming you want to convey information related to your title, a neutral assessment (ask a friend or wife or something if you need a another opinion if you don't believe me) is that MANY of your slides convey no useful information about quaternions nor why you believe they are "THE" numbers of the universe.

The experiment was conducted: I gave the presentation to two different people.  One pointed out a few flaws in the transitions between the slides near the start of the talk.  It is something I noticed while practicing the talk.  Otherwise, he was not generous in his assessment of you as an academic.

The other person said the following:

Whilst your slides on their own do not explain an awful lot - it is not reasonable to criticize them as they were merely an adjunct to your

lecture. Moreover I personally appreciate your easy going approach to the work you are doing.

Such are the limitations of ignite talks.

In contrast, by my count, you would have me drop more than half the slides.  I did the experiment, and have shown your advice on the presentation should not be used.

I wrote something in reply as I recall, which was totally ignored.

We are like a divorced couple.  I thought you granted the divorce a few blogs ago, but now you are just trying to help, offering a few good slides instead of those silly animations.

So lets talk about the animations you would gut.  That used to be the only physics I could share with my wife and daughter: isn't this movie fun?  I now have the updated versions of Z1, Z2, Z4, Z2xZ2, and Q8 they like.

You are of the opinion that the animations have no deep meaning.  Since you are a cynical critic, nothing I say will sway your opinion.  This is for other people reading our divorce proceedings, so they can decide who to remain friends with.

Here are three graphs of the complex plane.

Only the middle one could have anything to do with a reflection of the real or imaginary axis.  Since neither axes was labelled, it could be either.

Here are three frames from an animation like the one in the talk, with the initial input going from one vertex in a space-time cube, say {-5, -5, -5, -5} to the origin, and then including either a reflection over the real line or a reflection over the three imaginary axes:

The middle frame must be from a reflection of the imaginary axes.  The right frame must also be from a reflection of the imaginary axes as the time reflection has a point always on the screen.  The left panel is probably from a time reflection.  The one possible exception is when the mirror reflection animation is at the origin, so there is but one point.

The animation representation feels closer to the real world.  I don't need a labelled axis to tell the difference between a reflection in time and one in space.

So I addressed the biggest issue.  The smaller issues have to do with how to tell a story, one that can connect with people.  Golly, the world is full of rough things and studying that obvious subject that is voided in classical geometry lead to one of the biggest changes ushered in the last century.  Golly, the world is exclusively about stuff that doesn't last, no matter what time scale one picks.  Where is the transient math to match?

The music was in the last slide on a Quaternion Jam session.  In a Jam session, people show up to play together, make mistakes together, and hopefully find something that sounds better than anything they could do on their own.  I am far too aware of my own limitations to think I can do a project this size on my own.

The real experiment is less than 24 hours away.  I'll report on the outcome in the body of the blog when I get a chance.

"In contrast, by my count, you would have me drop more than half the slides. I did the experiment, and have shown your advice on the presentation should not be used."

Well, the idea wasn't just to drop slides and leave dead air. Obviously the point was to add more slides conveying information about why you felt "Quaternions [are] THE Numbers of the Universe".

I commend you for asking advice from friends (I really do), but you did the wrong experiment. Do your presentation, and then ask them questions regarding what you tried to convey. If it mismatches, explain (this is outside of the presentation now of course) so they now understand, and ask what you could do to help convey your point better in the presentation.

Otherwise you end up with:
"That used to be the only physics I could share with my wife and daughter: isn't this movie fun?"

That only assess whether the animation was amusing, not whether they got the education from it that you intended.

You waste an entire 20% !! of your talk on these animations. And you seem to have misunderstood the point of my objection to them. I see that if you change a sign in one of the components, that you (by tautology) get the quaternion back with that component with a changed sign. I could do this with ANY multicomponent entity. It tells nothing special about quaternions, and more importantly it doesn't explain WHY or HOW you feel this makes quaternions well suited for describing the universe.

And on top of that you insert insults to quaternions. You call them "a one trick pony" and even put an entire quote in insulting them. For your last post it's starting to sound like your goal is to wow people into possibly wanting to learn more latter and work with you. If your intent is to give a recruiting speech, talk about what makes quaternions special in your mind. You don't see campaign ads where a candidate points out reasons why you shouldn't vote for him. Your message is diluted.

"The real experiment is less than 24 hours away. I'll report on the outcome in the body of the blog when I get a chance."

Good luck. Hopefully I'm wrong and everyone leaves your talk inspired with whatever nugget was your intent.

I'm liking there Lorentz Boost in Quaternions, although I doubt I'll be able to remember where the
conjugates go.
Here is how I pull the trick off.  A rotation in 3D space is done with quaternions like so:
$\\ R \rightarrow R'=uRu^*\\ \\ \rm{where}\\ \\ u = \{\cos(\alpha), \vec{I} \sin(\alpha)\}$

Back in 1910 and 1911, folks tried to write an equation like this to do rotations in spacetime with real-valued quaternions.  They swapped in a hypercomplex cosine and hypercomplex sine.  That didn't work.  It gets two terms right, but two others wrong.  If they tossed in an extra factor of i that commutes, then for complex-valued quaternions, then one can create a nice, compact equation.

The game becomes one of how can one add in the right stuff, and subtract the wrong stuff, still using the same stuff?  So one way to do something different is to have the two trig functions right next to each other, first both not having a conjugate, then both having a conjugate.  I do need to memorize that both of these "on-the-same-side" triple product terms need a conjugate and are a difference over 2, but that is not too much info to memorize.

What is more interesting to me is thinking about the groups one needs for the 3D spatial rotation verse the one for the Lorentz boost.  The former is simply connect while the latter is not.
"What role these vector terms in the potential quaternion may play, if any, is unknown to me." B is really a bi-vector?

The magnetic field B is a bi-vector.
Dough

Why do you hardly ever use the fact that quaternionic distributions all show eight different sign flavors. This fact on its own is hardly known, but these sign flavors are still heavily used in physics. However, they are encoded in gamma matrices and spinors. The Dirac and Majorana equations use them.
Did you ever notice that a switch from a complex wave equation to a quaternionic wave equation changes linear equations of motion, such as the Dirac and Majorana equations into balance equations (continuity equations)?
Such a switch turns the wave view of physics into a flow view of physics. It does not change physical reality, but it certainly changes the view that this reality offers.

Hans
If you think, think twice
I have not been able to construct gamma matrices with quaternions.  I did get within 5 factors of i to gamma matrices using quaternion triple products.  The Dirac and Majorana equations are at the core of relativistic quantum mechanics.  For the audience at Ignite Boston, I think even LordKelvinForBreakfast would think that was too much.

I do have dreams of 4-parameter animations that make solutions to both Dirac and Majorana equations viewable.  That day is quite a ways away.

As an ultra-conservative fringe guy, I am not yet putting any money on the table to say I have an insite into the eight different sign flavors of QCD (if that is what you are referencing).  One would need in addition some concrete algebra about the game of quarks, particularly the +2/3, -1/3 charge business.  That is a long stretch, interesting, but beyond my reach.
Dough,

You might take a look at another blog on this website. (See: Quaternionic Versus Complex Probability Ampitude Distributions)
There the route from spinors and gamma matrices to quaternions is exploited and applied to the Dirac and Majorana equations. It transfers them both into two quaternionic equations that use quaternionic wave functions. Thus it transfers them into balance equations. It paves the road to a lot more of the elementary particles. I agree that this stuff is too complicated for Boston. But that was not the question.

For the first time it becomes clear that quaternions are essential to physics. Quaternion quantum physics makes sense!
Indeed quaternions are THE numbers of physics.

Hans
If you think, think twice

I've seen for quite some time that Doug and Hans were of a set.  My only concern was what would happen if they ever noticed one-another.

Well, I guess we shall now find out.  ;)

Hans commented on an early blog of mine, and I replied to one of his.  It looks like he is focusing on a different part of the world than I am, namely relativist quantum field theory.  The math universe of quaternions is big enough for both of us.
The problem with quaternions is that they do not fit Minkowski signature. The reason is that time and space get connected when an observer moves with respect to the observed object.

When eigenvalues and fields are concerned, quaternions do fit without any problem.

Quaternionic functions have very special properties. This becomes apparent when quaternionic probability amplitude distributions are concerned. They can be considered as a package of a charge density distribution and a current density distribution or as the combination of a scalar potential and a vector potential. They convert linear equations of motion into balance equations. This makes them very suitable for fluid dynamics problems.
With quaternionic probability amplitude distributions the Helmholtz decomposition theorem makes more sense.

The sign flavors of quaternionic functions open up new kinds of physics. This is shown by the quaternionic versions of the Dirac and Majorana equations.

Thus by separating relative displacements (observer-observed) from other dynamic features the discrepancy with spacetime can be solved and the actual value of quaternionic functions will come to the surface. Relative displacements can easily be treated separately.
If you think, think twice
I am having trouble with the first line:
The problem with quaternions is that they do not fit Minkowski signature.
Let's say I have 2 events, and calculate the difference in time and space between the two.  There are two distinct calculations I can do with this data:

$\\ (dt, d\vec{R}/c) \times (dt, d\vec{R}/c) = (dt^2 - dR^2/c^2, 2 ~dt ~d\vec{R}) = (d \tau^2, 2 ~ dt ~ d\vec{R}/c)\\ \\ (dt, d\vec{R}/c)^* \times (dt,d\vec{R}/c) = (dt^2 + dR^2/c^2, \vec{0}) = (||dq||, \vec{0})$

Two darn similar calculations, two familiar results.  One is about what intertial observers can agree about.  The other is a way to impose a least lower bound of zero on a quaternion.  I believe I have read (and reread because it didn't make much sense to me) that people hoped the norm of a quaternion would be the interval.  Apple, meet orange, they are two different things and it should be OK that there are two different things.

If this is the issue, please tell me more about why I should consider this a problem.  If this is not the issue, my bad.
My approach to spacetime is different, but it is founded on a quite different reasoning. Quantum logic and separable Hilbert spaces do not provide a means to represent time other than as a global parameter. This comes down to the fact that these mathematical constructs can only represent a static status quo of the universe. The fact that both the Schrödinger picture and the Heisenberg picture are valid representations of physics supports this conclusion. It means that if quantum logic or a Hilbert space must be used, then physics must be represented by a sequence of these constructs. That is why the corresponding model is called Hilbert Book Model. In this model each page represents a static status quo of the universe. The page counter plays the role of a progress parameter. Time only gets involved when an observer is treated that moves with respect to the observed item. It does introduce two notions of time; the proper time (=time of the observer) and the coordinate time (=time of the observed object). Both notions are related to the progress parameter. Together with location in space these notions of time constitute the notion of space time. Location can be connected to an operator and to the progression parameter. The eigenvalues of this operator can easily be connected with quaternions. The relation with spacetime is far more complicated. Spacetime is never an eigenvalue of an operator.

Thus in the Hilbert Book Model quaternions play a major role as eigenvalues while spacetime is merely introduced as an aftermath.
If you think, think twice
i have no ability to make a useful comment ..

but i can make a trivial one ..

.. i only examined (in a whimsical way) quaternions very recently. And now i'm annoyed that, (way back) at university, they didn't introduce us to quaternions prior to dealing with vector analysis. It always seemed to me that vector analysis was a hodge podge of stuff that wasn't as coherent as it ought to be (which made it more difficult to absorb than it should have been). Now i know why.

The basic view of quaternions that i got is that they're essentially complex numbers. But the "imaginary" part is 3-D instead of 1-D. So the square root of -1 is any point on a unit circle about zero in 3-d "imaginary" space. But by extending into real + 4-D you lose commutative multiplication. .. I think this can be extended to 8-D, but you lose some other math operational symmetry (i'm winging it here).

So, yeh.. quaternions! They Rool! 8-)

[/waffle]

For what it is worth, I don't think your comments are trivial.  The cross product sure looks like it was made up out of thin air.  There was a big battle over complex numbers. Those squabbles created the labels "real" and "imaginary" for the two players.  The label insult stuck.   There is nothing imaginary about imaginary numbers.  Complex numbers have proved useful, so they are still taught.

Quaternions got involved in a similar battle as complex numbers, but lost.  It is the engineering professor Gibbs at Yale who gets credit with the broad acceptance of the hodge podge known as vector analysis.  One shortcoming of my short presentation is I cannot say that all the things I do with quaternions could be accomplished by a clever combination of vector tools.  The fancy way to say it is they are the same up to an isomorphism.  One can write equations either way - a point not widely accepted since people don't think there is a way to do Lorentz boosts.  A point of using quaternions is that the clever part can be omitted.  Square a quaternion and voila, the first term plays an important role in physics.  To do this with vectors, I need to teach about a covariant tensor and a contravariant tensor and a metric tensor.  Sounds like more work to me than squaring.

Let me fill in some details in the winging it part.  Each step one takes from real numbers, a property is lost:

real -> complex numbers (2D)  no longer a totally ordered set
complex -> quaternions (4D)  multiplication no longer commutes
quaternions -> octonions (8D)  no longer associative

I am not sure what is lost going up to 16D.

Doug
A math discussion at work today made me think of you and I decided to see how things have been going here.  Skimming through your articles it looks like you eventually decided to (mostly) focus on taking the time to learn the subjects a bit more. That's wonderful and I wish you luck.
I am not sure what is lost going up to 16D.
Note also that the octonions no longer have a group structure since multiplication associativity is lost. Beyond the octonions, you no longer have a normed division algebra.  This means that two non-zero numbers can be multiplied to obtain a zero result.  So it loses enough structure that the sedenions (the 16D version you refer to) can't be used as a basis for a "number system".

A natural way to extend from real -> complex -> quaternion -> etc. is called the Cayley-Dickson construction.  From the very way it is defined, one cannot lose additive associativity or multiplicative distribution over addition.  So in a fairly real sense, there is 'nothing left to lose' after losing the last shred going from octonion to sedenion.  So you can continue beyond 16D but nothing all that interesting happens.

Ignite talks short, all of 20 slides, 15 seconds a slide, so 5 minutes total.
It is very ambitious to introduce any subject in such a short period of time.  Unfortunately I have to agree with the assessment that it doesn't seem to be very focused. You are an interesting person, so I have no doubt that you can entertain a crowd for 5 minutes.  But it would be best to decide what you want to convey and focus your talk to achieve this goal.

There was a big battle over complex numbers. Those squabbles created the labels "real" and "imaginary" for the two players.  The label insult stuck.   There is nothing imaginary about imaginary numbers.  Complex numbers have proved useful, so they are still taught.
I agree with the sentiment that complex numbers are just as "real" of mathematical objects as ordinary numbers, and furthermore that they have proved very useful.  Although I don't consider the label imaginary an insult.  Even in quantum mechanics which looks the cleanest formulated with complex numbers, the observables are all real valued.  When teaching electrodynamics it would often be useful to use complex numbers to help keep track of the phase of electromagnetic waves in an easy manner.  However, a common mistake was for students to get too comfortable with complex valued waves standing in for what was really real valued waves, and forget to reduce down to a real value before multiplying field intensities when calculating energy densities, momentum, etc.

Square a quaternion and voila, the first term plays an important role in physics.  To do this with vectors, I need to teach about a covariant tensor and a contravariant tensor and a metric tensor.  Sounds like more work to me than squaring.
And here is the old Doug coming out.  With all the diving into details of groups and subgroups, you are still way off the mark on the big picture.

You are essentially obsessed with numerology. You see a four here and a 4 there, which can further be grouped into (1,3) and (1,3) and you get all giddy.  You see patterns everywhere and equate them to deep truths that you wish to pull more from instead of just coincidences.  You are just chasing numerology.

You are not being intellectually honest with yourself.  You have already made up your mind on whether these coincidences have some deep connection between quaternions and 4-D spacetime, instead of asking probing questions from both sides to see what is going on.  You never ask why quaternions fail so miserably at this or that, you only ask can I shove them into this or that, and then decide if you did it that it meant the quaternion is wonderful and gives you a different view on it.  It doesn't.

For a comparison, consider actual geometric objects like vectors, tensors, scalar, and how often we can just "get" to the answer to physical questions by combining them the simplest way possible to get an object with the symmetries we require.  All your manipulations with quaternions instead involve you knowing the answer, and shoving quaternions to make them match the predetermined answer.  And even then you often accidentally ruin symmetries you wanted, because quaternions are so ill-fitted at representing the objects you are trying to model with them.

Consider modern vector analysis, which is just a subset of differential geometry.  In some sense, original vector analysis fell prey to some numerology issues as well.  In Euclidean space with 3 spatial dimensions, some relations naturally fall out that seem to have some similar properties... ie. interpreting the cross product as yielding a vector just because it has the same number of components, etc.  These don't generalize well outside of the "magical" 3 dimensions.  Latter when more was understood, and looking back through the eye of differential geometry, we can see the problem was people overly "intuitively equating" things just because they had some similar properties.  Unlike quaterions (which are geometrically just numbers, not vectors, no matter how much you try to make them something else), differential geometry isn't limited to four dimensional space, or non-curved spaces of the original vector analysis.

Are quaternions extra-well suited to treat spacetime? No.  The _best_ you can do is try to find the spacetime that maximizes the coincidences.  Throw out all spacetimes except four-dimensional ones.  Then throw out all spacetimes that are curved.  Then throw out everything that isn't Lorentzian.  And even then, throw out the cherished lesson of relativity that coordinate systems are arbitrary and restrict yourself to the coordinate basis of inertial frames to make the correspondence with the quaternion basis look stronger.  And then even define the 'length' of this quaterion not by the natural definition using the fact that its a number (it's magnitude) but multiply it by itself and extract a single element from the result.

That is so contrived, it is unclear how you can view it as simplifying. It is not.  Instead a better question would be why they appear similar at all, as maybe something _can_ be learned from that.  The reason is that this first component resembles the using the metric of flat Lorentzian spacetime in an inertial coordinate system basis to get a scalar from a vector.  To consider if this is a deep truth or just a numerological coincidence, just try deviating from that at all.  Any other coordinate system on the same flat space time? Nope, breaks the correspondance.  What about curved spacetime? Nope, quaternions don't give us useful insight in how to generalize to get there.  What about different dimensions? Nope, quaternions cannot be generalized in this fashion either.

That you scoff at four-vectors needing a metric to spit out a scalar, shows how blinded you are here.  That your "vector" operation spit out a number without any input on the geometry should be screaming at you that their combination in this manner therefore doesn't contain geometrical information ... unless of course you contrive it to.  You are taking a break to study and that is wonderful, but if you pursue studying only as a digging adventure to find more baubles that you can lay on the table and try to squint at to see patterns ... well, that is just 'fact collecting' for your narrow purposes and not true learning.
(EDIT: I'm not trying to kill creativity.  Have a hunch? See a similarity that makes you curious? Dig into it, but be intellectually honest about it otherwise you'll never be able to re-assess your initial assumptions and really learn from the journey.)
The construction formula of Warren Smith results up to the octonions in the same hyper-complex numbers however the sedions differ from the 2^4-ons. The 2^n-ons tend to have slightly better algorithmic properties than the hyper-complex numbers that are constructed via the Cayley Dickson construction formula.

Both construction formulas ignore the potential sign selections. A 2^n-on has 2^n independent sign selections. One of them is conjugation. The other are reflections. All quaternions and quaternionic distributions feature eight different sign selections.

The real power of quaternions does not locate in the numbers, but in the functions and distributions that can be constructed. For example quaternionic probability amplitude distributions (QPAD's) are very suitable for application in fluid dynamics.

It is hardly ever done yet, but these QPAD's are also very suitable for representing the state functions of quantum physical particles. However, this application involves two extra steps. It involves the interpretation of the QPAD as a combination of a charge density distribution and a current density distribution and the extra interpretation that these distributions describe the situation of what occurs in the shared parameter space of these QPAD's. These two steps revolutionize fundamental physics. They offer a completely different view of what happens in the lowest undercrofts of physics.

These powerful features widely compensate the difficulties that are introduced by the non-commutativity of the product rule of the quaternions.
If you think, think twice
Hello Henry:
Thanks for the info on sedenions.  It must be almost as bad as using the Klein 4-group to do physics to pick a flop that once was dear to me.

I have a track record of getting quaternions to do things that I have read they cannot do.  It is true I have yet to try to do extra dimensions, but I have to admit, I have zero interest in the sport.  My gut feeling it work in extra dimensions, while beautifully ingenious, is a way to keep wicked smart people employed, but will be a house of card that refuses to collapse even by the end of 2014 if the LHC shows there are no super particles around.  So the fact that we agree quaternions are ill suited for extra dimensions you consider it a bug, I consider it a feature.  That is a clear distinction we can both live with I hope.

Real numbers can be used in curved spacetime. Complex numbers can be used in curved spacetime,  Quaternions should be able to be used productively in curved spacetime.  I don't know how to do it.  I have said from the start that quaternions cannot be used for a field theory of gravity.  Gravity can be described by a dynamic metric theory.  Changes in a symmetric metric are symmetric.  Quaternions are asymmetric, and it is there anti-symmetric part that bars them from contributing to describing gravity.  Expressions written as quaternions can respond to gravity, but that is a different story.

I have on several occasioned complained about the rules for accounting with distances.  No matter what is measured, I want to use the same darn rules: fill in all four slots.  Nature is consistent, people gravitate to convenience.  If the other three numbers are super close to zero, go ahead, develop tools that toss information under the bus without pointing out it exists.  That is what the Einstein summation convention does.  The account in me is upset that dt dR doesn't even have a name.  Nature uses everything.  I don't expect you to accept that idea, but it is a pea under my pile of mattresses.

Doug
Doug
Be careful when you say where quaternions can be applied and where not. It is always possible to introduce a smooth quaternionic function from a quaternionic parameter. This immediately introduces a quaternionic manifold that will be curved. I used this idea in order to handle the shared parameter space of quaternionic wave functions. Because this is such a special space I wanted to give it a special name. The wife of a friend who specializes in antique history suggested the name Palestra. In Greek antiquity it is a public place that is used for training or exercise in wrestling or athletics. Thus the Hilbert Book Model uses the name Palestra for the space where everything in universe happens.
If you think, think twice
I have said from the start that quaternions cannot be used for a field theory of gravity.
What are you talking about?  You may have abandoned your many versions of your theories of gravity using a quaternion potential field, but you have quite the gall to claim with a straight face that you "have said from the start" that such a thing would never work.
Real numbers can be used in curved spacetime. Complex numbers can be used in curved spacetime,  Quaternions should be able to be used productively in curved spacetime.
Argh. It's like you have serious comprehension issues when it comes to certain subjects.  I've tried to explain the issue multiple times in the past and I see CuriousReader gave another try below.  If you don't understand, ask questions, but at least take the time to learn.  Don't just skip on by, pushing the misunderstanding forward to come up yet again.

I have on several occasioned complained about the rules for accounting with distances.  No matter what is measured, I want to use the same darn rules: fill in all four slots.
So anything involving vectors must result in another vector?  We can't have scalar quantities, or tensors?  In your obsession with "accounting", you are taking one rule you made up, and applying it out of context to everything.  That's not good accounting.
If the other three numbers are super close to zero, go ahead, develop tools that toss information under the bus without pointing out it exists.  That is what the Einstein summation convention does.  The account in me is upset that dt dR doesn't even have a name.  Nature uses everything.  I don't expect you to accept that idea, but it is a pea under my pile of mattresses.
You clearly don't even understand the Einstein summation convention.  When you calculate the magnitude of a spacetime vector, there is very much a dt dR piece.  It just so happens that in the specific case of inertial coordinates systems in flat spacetime, the metric component for dt dx, etc is zero. In other coordinate systems in may not be.  And ironically, in your attempt to call the spacetime magnitude the 'scalar' part of the result of multiplying two quaternions, it actually is you that is violating the accounting you are complaining about.
I have productive discussions with others because I don't face distortions of what I claim so often it makes me angry.  I will only deal with one point, the first one:
What are you talking about?  You may have abandoned your many versions of your theories of gravity using a quaternion potential field, but you have quite the gall to claim with a straight face that you "have said from the start" that such a thing would never work.
Before I started blogging here, I had printed up a t-shirt with my proposed Lagrange density on it.  Half the terms used were generated using the rules of quaternion multiplication.  The other half used the Klein 4-group rules.  The ones that used the rules for quaternion multiplication lead to EM.  That part remains correct to this day because it is isomorphic to the standard approach to the Maxwell equations.  It was only the terms that use the Klein 4-group rules that I hoped to connect to gravity.  As you and others pointed out, there were a number of problems with the proposal I called GEM which I retracted.  The gravity part never used a quaternion.  It used a 4-potential equipped with different rules of multiplication.  That is were my gall comes from, a familiarity with the details of my proposal.
Do you wish to expand on this comment?
I agree with the sentiment that complex numbers are just as "real" of mathematical objects as ordinary numbers, and furthermore that they have proved very useful.  Although I don't consider the label imaginary an insult.  Even in quantum mechanics which looks the cleanest formulated with complex numbers, the observables are all real valued.  When teaching electrodynamics it would often be useful to use complex numbers to help keep track of the phase of electromagnetic waves in an easy manner.  However, a common mistake was for students to get too comfortable with complex valued waves standing in for what was really real valued waves, and forget to reduce down to a real value before multiplying field intensities when calculating energy densities, momentum, etc.
Why do you think that is so?  There must be an underlying reason.  I know I have a solid reason, but would prefer to hear your response first.
Doug,

There are many well considered, informed -- and patient -- replies to your posts.

For example:

"... Are quaternions extra-well suited to treat spacetime? No. The _best_ you can do is try to find the spacetime that maximizes the coincidences. Throw out all spacetimes except four-dimensional ones. Then throw out all spacetimes that are curved. Then throw out everything that isn't Lorentzian. And even then, throw out the cherished lesson of relativity that coordinate systems are arbitrary and restrict yourself to the coordinate basis of inertial frames to make the correspondence with the quaternion basis look stronger. And then even define the 'length' of this quaterion not by the natural definition using the fact that its a number (it's magnitude) but multiply it by itself and extract a single element from the result.
..."

Here you tell us about your planned presentation ("No, you cannot go unless...") and ask for comments. Did you possibly think that this scattered confused error-filled silliness would be praised?

You seem to relentlessly disregard constructive information and facts you don't like. Your posts and your posturing replies to comments from folks who obviously know much more than you do are just cringe worthy.

Throw out all spacetimes except four-dimensional ones.
Guilty as charged. You can get a good paying job manufacturing this kind of stuff, but in the long run, I think all work with more than four-dimensions will be wrong.  Every.  Last.  Paper.
Then throw out all spacetimes that are curved.
I hope you are not nieve enough to accept this to be true.  I don't know how to cow rope quaternions to the standard machinery of curved spacetime, the details are arduos.  Quaternions are a [division algebra] There always must be a way from here to there.  In fact, there are always a bunch of different ways from here to there.  One way is to say that the basis vectors change depending one where one happens to be in the spacetime manifold.  I can picture how to program curved spacetime.  I don't know when or if I can implement the code, but it is easy to imagine generalizing the step for multiplication in a way that would look like curved spacetime.

If you go to the literature, it is really clear: there is not a way to derive the Maxwell equations using real-valued quaternions.  It is there, published by folks smarter than I.  There is no way to represent the Lorentz group using quaternions.

"Constructive criticism"? Surely you jest one-who-cringes.  Ed has imposed chains, claiming I can only ever work with inertial reference frames.  Might as well give up hope of doing anything for gravity if those are the rules.

What I expect was the title to draw out all the supporters of Lord Kelvin to stand up and be counted.

Doug:

You state:

... Quaternions are a mathematical field. ...

Now, you didn't really think you were going to get away with this misstatement, did you?  The Quaternions are not a mathematical Field, because multiplication is not commutative.

The Quaternions are a Skew Field AKA a Division Ring.

Come on, we've told you this before.  ;)

David

I was in a rush between getting out of work, and going to give a talk.  My bad, corrected.  One can use real numbers in curved spacetime, and complex numbers in curved spacetime, so I see no reason not to let quaternions be used in curved spacetime.
This quik synopsis from Wiki really sez it all, wrt quaternions. Hamilton was a great mathematician & physicist, but in the 130 yrs since he discovered them, physicists have deferred to vector & tensor analysis, & no one has established a permanent niche in physics for quaternions.
'From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature of quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics.'
I don't see that changing anytime soon.

One would need a reason for such a change.  Right now there isn't one.
The fact that the "other aspects of quaternions" were not noticed is a shame for last century's scientists. That does not say that these other aspects have no significant impact. Wiki's article also neglects these aspects. The reason is the notability procedure that Wikipedia applies to new contributions. You will never find fresh scientific ideas in Wikipedia. For new ideas Google is a better resource than Wikipedia.

The "other aspects" are mentioned in comments above.
If you think, think twice
You wrote:
"One can use real numbers in curved spacetime, and complex numbers in curved spacetime, so I see no reason not to let quaternions be used in curved spacetime."

You repeated this a couple times in the discussion above.
Lately you have been learning a lot I think from David, showing clearly your ability to learn from others. But there seem to be some topics that are sacrosanct to you, and as soon as they come up you show this amazing ability to blind yourself instead of really listening and learning. The above quote from you seems to be displaying this particularly strongly. I think it might be useful for you to read through some of the discussion again.

If you have some property you want to describe with a __NUMBER__ then it doesn't matter the dimensions or geometry of space(time). Imagine the simplest case: there is some property defined at every location. If the property it just a single number, you could represent this as a function: (location) -> (number). Nothing in this general description restricts the type of number you use.

However, the problem is you have this habit of not treating quaternions as numbers. You seem to want to treat them as all kinds of objects, and mix and match them as such. Most importantly, is you keep mentioning that if one considers a quaternion as a collection of a time piece and three spatial pieces, that multiplying two quaternions gives you what looks like (in the time position of the resulting quaternion) the spacetime dot product of two four-vectors described with an inertial coordinate system in Minkowski spacetime. Basically, as soon as you want to extract and use this fact to use quaternions as four-vectors, you need to be hyper-aware of what requirements you are placing on yourself to make such an analogy. For some reason, every time someone brings this up, you either dismiss them, misunderstand them, or just incorrectly claim they are wrong.

And again, as has been brought up before, you don't even need to go to curved spacetime to break the relationship you are trying to exploit. Just consider non-inertial coordinate systems in Minkowski spacetime. The multiplication of quaternions is a built in part of them ... you can't change their multiplication (their group structure if you will). So to make the association of (quaternion multiplication --> spacetime dot product) as you are doing, breaks as soon as you go away from the one specific case that aligns this coincidence as greatly as possible. That is because the metric, written out in components, changes from the diagonal 1,-1,-1,-1 that you are requiring to make this associationg.

So please take the time to listen and learn.
David seems to be able to reach out to you better than anyone else, so maybe he can give a nice quick summary explanation to help you (David: I'd also appreciate any guidance you might have, as Henry and others may be overstating somethings ... but the dot product issue so fundamental to your claims, seems unavoidable).

The discussion here unfortunately led me to staying up too late reading on quaternions, octonians, etc.

Here's an interesting article discussing the history behind the quaternion confusion
www.lrcphysics.com/storage/documents/Hamilton%20Rodrigues%20and%20Quater...
I wasn't aware that there was no such thing as a vector at the time, and Hamilton actually created the name for quaternions.
I haven't finished reading it, but so far it seems to say one serious problem was along the lines of what people above said -- trying to treat a number as a vector and not being careful about what is required to make this correspondence. Since the concept of a (modern) vector wasn't understood then, you can hardly blame Hamilton for conflating some of the concepts.

Thanks for the reference which I got a chance to read. It is not my practice to dive into the tangled history. The story told was centered on what I referred to as the "one trick pony", how to properly do 3D rotations. The possibility of doing what I called "the math of transience" using animations was of course not address there, nor by the parade of critics here except LordKelvinForBreakfast who wants nothing of it. So it goes.
i notice (from my uneducated point of view) that quaternions are usually dealt with in the form:

real + imaginary, where imaginary is i,j,k 3D

is there ever any advantage to seeing quaternions extending 2D complex from:

real1 + i*real2

to

complex1 + i* complex2 .. where complex = x + j*y, and i*j = k

?

[/random, naive question]

In fact this is the base of the Cayley-Dickson and Warren Smith construction formulas for building higher dimension hyper complex numbers from lower dimension (hyper) complex numbers.

Warning:  these construction formulas neglect sign selections.
If you think, think twice
The videos are up, but not at the links Doug gave.

Here is Doug's talk: