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Quaternions: an old (and new) look at four dimensions - page 2

THE  STORY  SO  FAR

In page 1 , the real number system was introduced from the point of view of a school student, gradually building from the natural numbers (1, 2, 3,...) to include zero, fractions, negative numbers, and irrational numbers (both surd and transcendental). This led to the concept of a real number line - and ultimately, the Cartesian plane, in which graphs showing relationships between real numbers could be plotted.

Also discussed were the extension of the real numbers to the complex numbers, and the corresponding modification of the Cartesian plane to the Argand plane. The fact that multiplication of complex numbers can be shown to have a strong link to the idea of rotation was illustrated.

Another aspect of both real and complex numbers which was mentioned is the fact that each of these sets of numbers forms a field. This basically means that they obey certain rules or "laws" which essentially guarantee that their arithmetic is "well-behaved", and doesn't hold any nasty surprises.

The stage is now set for the entry of quaternions.

In the 1830's, Sir William Rowan Hamilton (1805-1865), the Astronomer Royal of Ireland, became interested in trying to extend the notion of complex numbers to three dimensions. Expressing complex numbers as "couplets", or ordered pairs [so that 5-7i might be expressed as (5,-7)], he wanted to produce a system of "triplets" on which one could perform arithmetic.

The idea was to have a second imaginary unit, j, which would square to give -1 (just as i does), and which could be multiplied by i to produce meaningful results. Adding and subtracting triplets presented no problem; but he could not find a way to multiply them without introducing inconsistencies.

On 16th October, 1843, he had a flash of inspiration. Triplet multiplication would never work; but if two new imaginaries, j and k, were both introduced - giving three imaginaries altogether - then you could produce a system which would be internally consistent.

He was out walking with his wife when this dawned on him; and, on an impulse, he carved into the stonework of Broome Bridge, Dublin, the following cryptic inscription:

i² = j² = k² = ijk = -1

These days, you'd probably be arrested for damage to property if you did something like that; however, not only did Hamilton "get away with it", but it is recognized as a famous historical event (in mathematical circles, at least); and in 1958, the Royal Irish Academy erected a plaque commemorating the incident! (You can see a photograph of the plaque by clicking on a link within the web-page mentioned above; also, click here for a rather amusing page with some photos of the bridge - with its plaque - and a bunch of mathematics/physics types having their picture taken next to it.)

So what does it all mean?

This statement is Hamilton's way of defining what he called "quaternions" by setting up some equations which govern how they interact. The i, j, and k are all "imaginary quantities", just as i is in the context of complex numbers. Each of them squares to give -1 ; having by now become acquainted with complex numbers, it's not too hard to accept this idea as being "reasonable". But what are we to make of the reference to ijk ?

Well, we can take "i² = -1" as a definition ("i is something which, when squared, gives -1"). Similarly, "j² = -1" and "k² = -1" are also definitions. Therefore, "ijk = -1" is another definition ("when you multiply i, j, and k, the result is -1").

Think of these four definitions as "building blocks". We can play with them, see how they interact with each other, and try to build something that looks like a stable structure with them. If real numbers "pop up" at any point, we'll allow ourselves to treat them as usual - in particular, allowing that -1 × -1 = 1 ; and also assuming the Law of Multiplication by Unity or LMU (from the previous page ), which states that multiplying by 1 doesn't change anything . (This may be thought of as an "experimental" approach to mathematics.)

What happens when we do?

For a start, let's try multiplying i by ijk:

ijk = -1 , therefore i × ijk = i × -1 , therefore i²jk = i × -1 , therefore -1 × jk = i × -1

Taking a deep breath, let's see what happens if we allow ourselves to write -1 × jk as -jk and i × -1 as -i , and then follow our instincts:

-jk = -i , therefore jk = i

Well, we're still here, aren't we? The universe didn't disappear in a puff of smoke; so whatever we just did can't be too bad - and it just might come in useful (whatever that means).

Let's do something similar, this time multiplying ijk by k; and this time we'll be a bit less coy about dealing with minus signs if they crop up.

ijk = -1 , therefore ijk × k = -k , therefore ijk² = -k , therefore

-ij = -k , therefore ij = k

So far, so good. By starting with ijk = -1, we've now managed to simplify both ij and jk . But how will we simplify ik?

Still being cautious, we notice that there's a "j" stuck between the "i" and the "k" in ijk = -1 . So let's try something different. Since we now have expressions for both i and k in terms of other symbols, let's just multiply them together and see what happens:

k × i = ij × jk = i × j² × k = i × -1 × k = -ik ,

therefore ki = -ik , or, equivalently, ik = -ki

That result doesn't contain a "j" , but it does tell us something very important - that, since ki and ik are different, we were right to be cautious about how we multiplied these things together. We've found an example which shows that, in general, the Commutative Law of Multiplication (CLM) doesn't hold for quaternions!

(Also, as we'll soon see, that last result we derived will turn out to be of practical use in its own right as we proceed with our analysis.)

We just showed that jk = i . Let's multiply both sides of this equation by k, so as to "kill off" the k. Noting that we can't expect the commutative law to hold, we multiply both expressions by k on the right:

jk × k = i × k , therefore j × -1 = ik , or -j = ik .

Since we know that ik = -ki , we therefore have

-j = -ki , therefore ki = j .

Note that we could also have obtained this result via a different route by starting with ij = k , and multiplying both sides of this equation by i, so as to "kill off" the i. This time, we'd need to multiply both expressions by i on the left:

i × ij = i × k , therefore -1 × j = ik , therefore -j = ik

as before, thus again concluding that ki = j .

So, at the end of all that, we have the following results:

ij = k ; jk = i ; ki = j ; and also ik = -ki , so that ik = -j

There's just a bit more to do. We'd be very surprised, at this point, if the two statements ji = -k and kj = -i weren't true. However, we can't assume anything; we have to check:

Combining jk = i and ki = j, we obtain jk × ki = i × j ,

therefore jk²i = k , therefore -ji = k, or ji = -k

and similarly:

Combining ki = j and ij = k, we obtain ki × ij = j × k ,

therefore ki²j = i , therefore -kj = i, or kj = -i

YAY! So now we have it all. A final summary:

ij = k ; jk = i ; ki = j ; ji = -k ; kj = -i ; and ik = -j

There is a pleasing symmetry about all this. We've assumed only Hamilton's definition (originally a bit of graffiti on the Broome Bridge) and some basic, very reasonable properties of real numbers: that the CLM applies when multiplying reals by reals (just as in "usual" arithmetic); and that it also applies when multiplying a real number by one of these "imaginary" quaternions - both of which, it would seem, Hamilton assumed as "axioms", or "self-evident truths".

The amazing thing is, however, that the CLM is false when multiplying "imaginary" quaternions by each other! In these cases, the order of multiplication does matter.

Before moving on: a few comments about what we've just done.

Firstly: the kind of very careful, even pedantic, reasoning in which we've just engaged is typical of what mathematicians do. Manipulation of strings of symbols in this fashion is de rigueur in the branch of mathematics called "abstract algebra". (In the previous page, I did promise to mention something about "algebra" at some stage; now you have it.)

Secondly: if you read most abstract algebra textbooks, you'll probably find that the quaternions are not derived in this experimental way. You're far more likely to find that our conclusions above are presented as definitions. Why? Well, at the risk of stepping on a few toes, I suggest that the authors like to appear to be completely in control, and not allow of any suggestion of playful experimentation, of feeling their way toward a result - because it looks less than "rigorous" to do so. The algebraic reasoning will be presented, sure (and probably more tersely than I've done here) - but only after strict assumptions have been stated.

I take issue with this! Mathematics has not been handed down to us on stone tablets! It takes hard work and imagination to come up with a significant mathematical result. Clearly, this was true in Hamilton's case. (A more modern example is the proof by Andrew Wiles of Fermat's Last Theorem in the mid-1990's - it wasn't plain sailing!) I firmly believe that there needs to be far more motivation included in the teaching of mathematics; if this means recounting some of its history (warts and all) so that students realize that top-notch mathematicians are also human, and thus feel less intimidated, then so be it!

[The 19th century mathematician Carl Friedrich Gauss is considered to be one of the greatest mathematical minds ever; but in my extremely humble opinion, he has a lot to answer for. Brilliant he may have been, but his style was intimidating. (Have a look at this web-page to see what I mean. Also, visit this page.) If my website can go at least some of the way towards making mathematics more accessible to the lay-person - and less intimidating to the poor student - I reckon I've done something worthwhile.]

A similar situation exists in science. Ideas often take years or decades - or longer - to come to full fruition, and usually require the input of many scientists, all contributing, modifying and adding to work that has been done previously. For a classic example, see my electromagnetism page. (Again, I'd like to see more intellectual honesty about this - and related matters - in many physics textbooks.)

Some comments about the subject matter itself are also in order, before proceeding.

What we've done so far is to investigate the way multiplication works for quaternions - and, so far, only for the eight simplest quaternions: i, j, k, -i, -j, -k, 1, and -1.

That's certainly the most novel thing about quaternions; however, quite clearly, Hamilton was interested in adding and subtracting them as well - so that things like i+2j, 7-2i+j-5k, -j+4k etc. are included. Then multiplication among these would work as illustrated by the following example:

(i+2j) × (-j+4k) = -ij+4ik-2j²+8jk = -k-4j+2+8i , or 2+8i-4j-k

Thus, just as the set of complex numbers contains not just the four simplest elements (i, -i, 1, and -1) but also every possible expression of the form x + iy, where x and y are any real numbers - so also, the full set of quaternions consists of all possible expressions of the form

w + xi + yj + zk

where w, x, y, and z are real numbers; and the expressions can be added and subtracted, as well as multiplied and "divided".

It's worth mentioning, however, that the small set containing only i, j, k, -i, -j, -k, 1, and -1 is also important in its own right. Together with multiplication as just described, this set forms what is known as the quaternion group, which is of particular interest to mathematicians because it has certain special properties.

Since a quaternion can be composed of up to four components added together, multiplying them can be a bit of a handful. Whereas two complex numbers (which are each composed of a maximum of only two components) multiply together to give a maximum of four summands, with quaternions up to 16 summands are possible! So there's quite a bit of "donkey-work" involved if you're going to multiply them "by hand".

The quaternions, with addition and multiplication, obey all the field laws except one: the CLM. (When working with them, it's essential to bear this in mind to avoid silly mistakes.) In particular, the CLA does hold (thank goodness! ).

An "almost-field" of this type is sometimes referred to as a "division ring" or a "skew field". Unfortunately, both these terms include fields as well; perhaps "strictly skew field" may be a useful term. (This is one of those silly accidents of history - a bit like how we have to "agree" that electric current flows from positive to negative, even though we now know that the electrons which usually make up electric currents actually flow in precisely the opposite direction. In both cases, because somebody didn't clarify a situation in the early stages of development, misunderstandings and confusion have become entrenched.)

Another thing that's worth a quick mention is the Distributive Law, which deals with how addition and multiplication "interact" with each other. With fields (eg. the real or complex numbers), it's enough to give the "left" DL:  a(b+c) = ab + ac , because with the CLM working, we can then deduce the "right" version of the law:  (b+c)a = ba + ca . However, in a strictly skew field (such as the quaternions), it's necessary to spell out the left and right distributive laws, both of which are required but can't be derived from each other in the absence of the CLM. (Just thought you'd be fascinated... )

The word "division" in the paragraph before last brings up an important matter: how do you divide one quaternion by another? (So far, as you may have noticed, I've skirted around this issue.)

An approach somewhat similar to that for dividing complex numbers can be used. Fortunately, every quaternion has a quaternion conjugate. When a quaternion and its conjugate are multiplied together, the result is a pure real number (with no i's, j's, or k's). Also fortunately, the order in which they are multiplied doesn't matter - you get the same result either way.

The conjugate of w+ix+jy+kz is w-ix-jy-kz . Multiplying them together (in either direction) yields w²+x²+y²+z². This is strongly analogous to the way complex conjugates behave, and easy to remember. (As in complex numbers, the square root of this number is called the modulus of the quaternion, and may be thought of as its "distance" from the origin, applying Pythagoras's Theorem in a four dimensional setting.)

This means that the multiplicative inverse (or reciprocal) of w+ix+jy+zk is (w-ix-yj-kz)/(w²+x²+y²+z²), because if you multiply this by w+ix+yj+zk, you will get 1. (This is another of those cases in which the order of multiplication makes no difference.)

So, to "divide" one quaternion "by" another, we can multiply both by the conjugate of the divisor and then (if necessary) simplify the result.

Again, we have to bear in mind that the CLM cannot be assumed. Thus, if A and B are two quaternions, and if the multiplicative inverse of B is represented by B', in general A×B' will not give the same result as B'×A (which means that expressions like A/B are ambiguous, and should be avoided). So, again, we have to be careful to get the order right when doing the multiplication!

For example:

Let A = 2 + k , and B = 2 - i + j - 2k . (We'll work out both A×B' and B'×A.)

First: B' = (2 + i - j + 2k) / [2² + (-1)² + 1² + (-2)²] = (2 + i - j + 2k) / 10 .

Then:

A×B' = (2 + k) × (2 + i - j + 2k) / 10
= (4 + 2i - 2j + 4k + 2k + j + i - 2) / 10
= (2 + 3i - j + 6k) / 10
= 0.2 + 0.3i - 0.1j + 0.6k

and:

B'×A = (2 + i - j + 2k) / 10 × (2 + k)
= (4 + 2k + 2i - j - 2j - i + 4k - 2) / 10
= (2 + i - 3j + 6k) / 10
= 0.2 + 0.1i - 0.3j + 0.6k

- and, as you can see, they're not the same.

Note: if the square root of (w² + x² + y² + z²) = 1 , then w + ix + yj + zk is called a unit quaternion, as its "length" (or norm, denoted by ||w + xi + yj + kz|| ) = 1 .

As you've probably realized by now, performing arithmetic on quaternions "by hand" is a messy, complicated business - and you really have to concentrate to avoid making errors. It's not exactly difficult; it's just tedious. This sort of thing is done far better by computers than by people.

Of course, in the 19th century, when all this stuff first appeared, there weren't any computers. So it's not surprising that quaternions were viewed as something of a mixed blessing.

On the one hand, they were a novel, interesting phenomenon - especially back then, when a number system which violated the CLM was unheard-of. On the other hand, they're a pain to deal with, in practical terms!

Of course, this all begs the question: why would anybody want to be bothered with quaternions at all, other than for their novelty value?

After all, it's not easy to visualize what's going on with them. They're a four-dimensional system, with one "real" and three "imaginary" dimensions; and very few people claim to be able to get a mental picture of what happens in any more than three dimensions.

Two dimensions aren't a problem. The Cartesian plane, with its two "real" dimensions, and the Argand plane, with one "real" dimension and one "imaginary" dimension, are easy: you can draw detailed diagrams on a sheet of paper, etc. - but four dimensions present quite a challenge.

So is there any practical use for them? Is there any way we can apply quaternions to normal three-dimensional space?

Indeed there is. Computers are now everywhere, and don't share the problem that humans have when dealing with long-winded messy calculations; and one of the really neat things about quaternions is that they can be applied to computer animation, as I mentioned in the introduction to page 1 of this article.

Following is a bit more background about quaternions, and some links from which I hope you will gain at least some idea of how they can be applied to rotation.

Within the set of real numbers, each positive number has two square roots: one positive and one negative. The square roots of 1 are ±1, i.e. "plus or minus 1". Zero has one square root (itself); and negative numbers don't have any.

Within the set of complex numbers, every non-zero number has two square roots. (Zero still has one, as before: itself.)

Note that ±1 and ±i are all on a circle of radius 1, centred on the origin. The two square roots of 1 are still ±1. This time, -1 has two square roots also: ±i.

What about the quaternions?

In particular, we know that i², j², and k² are all equal to -1. In a similar way to -i in the complex case, it's easy to show that the squares of -i, -j, and -k are also equal to -1 in the quaternion case.

Is that it? Are there any more quaternion square roots of -1?

Let's consider what happens if we square xi + yj + zk (with at least one of x, y, or z not equal to zero):

(xi + yj + zk)² = (xi + yj + zk) × (xi + yj + zk)
= -x² + xyk - xzj -xyk -y² + yzi +xzj -yzi -z²
= -(x² + y² + z²)

How about that? It's a negative real number!

If we divide (xi + yj + zk) by the square root of (x² + y² + z²) , we obtain a quaternion which will square to give -1, whatever the values of x, y, and z may be. So we have a system in which -1 has infinitely many square roots!

What's more, these quaternions are purely imaginary, with no real component. Thus we have something which looks three-dimensional - something we can visualize. In fact, these square roots of -1 form a spherical shell of radius 1 in a three-dimensional space in which we can show numbers of the type "qi" along an x-axis, numbers of the type "rj" along a y-axis, and numbers of the type "sk" along a z-axis, where q, r, and s are all real numbers and q² + r² + s² = 1.

Of course, if q, r, and s have other (real) values, we get a differently-sized spherical shell, the radius of which will be the square root of q² + r² + s², because of Pythagoras's Theorem in three dimensions. Thus we can choose any position in space a fixed distance M (for modulus) from the origin.

Note: most quaternions have only two square roots. (As usual, zero has just one: itself). It's only negative real numbers that have infinitely many (quaternion) square roots.

(qi + rj + sk)² = -(q² + r² + s²)

The day after Hamilton discovered quaternions, he wrote a letter to his friend John T. Graves, in which he introduced the subject and also discussed square roots. (Click here to see my own page dealing with this matter further.)

Now, having given some preliminaries, I'll admit to not having a full understanding of how the rotation process works; so, rather than jump in feet first and make a monkey of myself, I'll refer you to some other web-pages where you can follow this up if you'd like to:

This link gives a very brief introduction to how quaternions can be used to show rotation.

Or, visit this web-page and scroll down to the bottom to find a link to a 10-page .pdf file which gives a more detailed introduction to this quite deep subject, but does so in as gentle a "beginner-friendly" fashion as you'll probably find anywhere.

As always, I recommend right-clicking to download this file to hard disc and then scanning it for viruses before opening it. It's probably OK; but with the possibility of "nasties" affecting internet traffic, it's better to be safe than sorry.

In these links, it is pointed out that the real part of a quaternion can be considered to be a scalar quantity, and that the three imaginary parts taken together can be viewed as a vector quantity. These terms will become important in the discussion of the final topic in this page, coming up next.

Using quaternions is not the only way to perform rotation. There are other methods - and different methods have their champions! In fact, there is some rather heated debate going on about the matter. To gain some idea of its intensity, and learn something about the issues involved, have a look at this web-page.

It must be admitted that this discussion of the rotational aspect of quaternions has become somewhat technical. Don't worry if you didn't follow it completely; unless you're thinking of going into computer graphics, you'll probably never need to! I just included it to give an idea of the flavour of the subject.

UPDATE, Friday, 21st June 2019 (winter solstice day, down here in the southern hemisphere!)

It's been a while since I've revisited this page to see which (if any) of the external links still work; it seems that at least some of them don't, unfortunately. However, someone has sent me a link to a page which does work - so far, at least! - and which contains some good general information about quaternions (thanks, Jim!). Well worth a look; here's the link:

https://www.mauriciopoppe.com/notes/mathematics/numeral-systems/quaternions/

I'd like to conclude this page with another aspect of quaternions. I'll admit that I don't fully understand everything about this either; but I believe that it is of such fundamental importance that some reference to it must be made. Indeed, I'll admit freely that it's the main reason I included these pages about quaternions at all.

Please note: in order to cut a long story as short as possible, I'll just give the "bare bones", and include several links to other web-pages. Please click on them and read their content to fill in the details.

As mentioned in that last link given above, quaternions caught many people's imagination in the mid-19th century. Hamilton believed that they would be of major importance in the development of mechanics; for a time, the idea was popular. Perhaps the phenomenon is similar to the way chaos theory and fractals "caught on" in the late 20th century (see my Mathematically-based computer graphics page for more about this).

However, as hinted earlier, in the late 1800's the popluarity of quaternions began to wane.

When James Clerk Maxwell (1831-1879) published his electromagnetism equations in 1873, he made a point of stating them in quaternion form.

In this link (about Hamilton - the same link given near the top of this page), toward the bottom, you can read how physicist William Thomson (Lord Kelvin) and mathematician Arthur Cayley had strong reservations about quaternions.

This link takes up the story of how, after Maxwell's death, J. Willard Gibbs and Oliver Heaviside did some drastic surgery on Maxwell's equations, effectively removing any quaternion content from them.

This link gives a brief introduction to the work of Tom Bearden, who has pioneered the suggestion that the removal by Gibbs and Heaviside of the scalar component of the quaternion version of Maxwell's equations, leaving only the three-dimensional vector component, has caused a great deal of damage, slowing scientific progress down dramatically over the last century.

This link is part of the same website as the page just mentioned, and is a link within that page. It gives more detail regarding the importance of restoring quaternions to their former place in electromagnetic theory.

This link is part of Tom Bearden's own website, and perhaps as good as any place within it to start. (His pages tend to be long and technical; this seems to be one of the most accessible to the novice in these matters.)

This article by Tom Bearden gives more background on these matters, including comments about Oliver Heaviside's difficulties with quaternions. A bit on the heavy side (pun intended! ), as is most of Tom's writing, but quite fascinating if you persevere with it.

Finally, a link to my Zero-point energy page, in which I pursue this further, and include links to pages dealing with Tom Bearden's Motionless Electromagnetic Generator (MEG).

With our world in deep trouble, this is an issue that needs to be known about and investigated fearlessly by intelligent, caring people - and, yes, I do count myself among them! If you do too, I encourage you to get involved. The matter is urgent.

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