Mad Teddy's web-pages

Quaternions: an old (and new) look at four dimensions - page 3

FURTHER  QUATERNION  STUFF

(and a brief look at other number systems with even more dimensions)

Page 1 of this set of three web-pages presented a background to the real number system that we all learned in school, and then progressed to some of the basic properties of complex numbers. The fact that both real and complex numbers form fields was mentioned, along with the fact that the multiplication of complex numbers can be seen to have a natural connection with rotation in the Argand plane.

Page 2 recounted the story of the discovery, in 1843, of the quaternions by Sir William Rowan Hamilton - and then went on to give a reasonably in-depth discussion of the basics how they "work", with particular emphasis on the fact that multiplication is not commutative, so that they form a (strictly) skew field. (I also growled about how I think that the over-formal way in which mathematics and science are often presented has caused a lot of damage from an educational point of view. )

The application of quaternions to computer animation was mentioned, along with the fairly recent suggestion that restoring the original quaternionic nature of Maxwell's electromagnetism equations may well have some extremely significant consequences for technology (and a whole lot more besides). In this connection, the concept of vectors was mentioned very briefly.

Along the way, the surprising fact that (in quaternion terms) negative real numbers have infinitely many square roots was raised. Hamilton's letter to his friend John T. Graves was mentioned in this connection.

In this, the third page, I'll develop the matter of square roots further (just because it is so intriguing, basically!), and briefly introduce the last of the four "normed division algebras" (the octonions), and also something called the biquaternions! - and maybe raise a bit of hell about a few other things too, while we're at it.

SQUARE ROOTS

Toward the end of Hamilton's letter to John Graves (mentioned above), he gives a formula involving a square root. It's not clear (at least, not to me) how he obtained this; but at some point I decided to try to work out, from first principles, my own general expression for the square root of a quaternion - and that formula appeared in my result.

I'll give my working shortly; but first, by way of motivation (and I'm sure that by now you know that I'm big on motivation), let's look at the simpler problem of finding a general formula for the square root of a complex number.

SQUARE ROOT OF A COMPLEX NUMBER

The aim of the following exercise is to find a general formula for the square root of the general complex number, x + yi , where x and y are both real.

Note: we can always put a complex number into its polar form (see page 2), and find its "principal" square root by taking the square root of its modulus and halving its argument; but here, we'll concentrate on doing it algebraically.

Suppose that p + qi , where p and q are also both real, is such a square root. Then

(p + qi)² = x + yi ,    therefore     p² + 2pqi - q² = x + yi

Equating real coefficients and imaginary coefficients, we obtain

p² - q² = x    and    2pq = y

From the second of these, we obtain q = y/2p . Substituting this into the first:

p² - y²/4p² = x ,    therefore    4p⁴ - 4p²x - y² = 0

This can be considered as a quadratic equation in p². Quartic equations of this type, in which linear and cubic terms are absent, are sometimes called biquadratic. It's possible to solve them by using the quadratic formula as a first step, as follows:

p² = [4x ± sqrt(16x² + 16y²)]/8 = [x ± sqrt(x² + y²)]/2

Note that we are using "sqrt" as an abbreviation for "square root of".

Now, since we are insisting that p must be real, and since [x - sqrt(x² + y²)] must be negative (unless y = 0, in which case the value of this expression will also be 0), we can simply drop the "-" from the "±". Then, taking square roots:

p = ±sqrt[x + sqrt(x² + y²)] / sqrt(2)    or    p = ±sqrt[sqrt(x² + y²) + x] / sqrt(2)

Next, substitute this into q = y/2p :

q = y × sqrt(2) / {±2 × sqrt[sqrt(x² + y²) + x]}

= ±y / {sqrt[sqrt(x² + y²) + x] × sqrt(2)}

Multiplying both the top and bottom of this fraction by sqrt[sqrt(x² + y²) - x] :

q = ±y × sqrt[sqrt(x² + y²) - x] / {sqrt[x² + y² - x²] × sqrt(2)}

= ±y × sqrt[sqrt(x² + y²) - x] / {sqrt[y²] × sqrt(2)}

= ±y × sqrt[sqrt(x² + y²) - x] / {y × sqrt(2)}

STOP RIGHT THERE!

That last step is not quite correct. Can you see why?

When we take the square root of y², it's reasonable to expect that the answer will be y. What else would it be?

When we talk about the "square root" of a (positive) number, it's understood that we're referring to the positive square root. For example, the "square root of 49" is 7, rather than -7 (even though they are both square roots of 49, since the square of each is 49).

If we were to use this process in a particular case with actual numbers for the coefficients, there'd be no confusion. However, when we use algebraic symbols (pronumerals) to stand for coefficients in fully-general formulas, as in our case here, we have to be careful.

In the development of our square root formula, y could be positive or negative (or zero). If it's negative, we need to be aware that the positive square root of y² is -y, not y.

To cut a long story as short as possible, and basically to "make things work", what we can do here is put y in the denominator - as we've done - but then also multiply the entire term by the "signum function", sgn(y). This takes the value  1  if y is positive,  -1  if it's negative, and  0  if it's zero. This fixes everything so that the formula will give us correct answers, no matter what the value of y is.

Okay - let's rewrite that last line with the corrrection, and continue:

... = ±y × sqrt[sqrt(x² + y²) - x] / {y × sqrt(2)} × sgn(y)

therefore    q = ±sqrt[sqrt(x² + y²) - x] / sqrt(2) × sgn(y)

So, finally, the two square roots of  x + yi  are

±{sqrt[sqrt(x² + y²) + x] + sqrt[sqrt(x² + y²) - x]×sgn(y) i} / sqrt(2)

Well, it's a bit tedious getting there; but the final formula itself isn't really all that complicated - and it does work! For example, it gives the square roots of 3 + 4i as ±(2 + i) . [You can check this by calculating (2 + i)².] Also, significantly, it gives the square roots of 3 - 4i as ±(2 - i), rather than as ±(2 + i), which it would give if we didn't include the sgn(y) in the expression!

Make up some examples of your own: choose some complex number, square it, and then use this formula to find the square roots of the result. You'll find that it always gives correct answers.

Note that the two square roots of any complex number are opposite each other with respect to zero in the Argand plane, corresponding to an anticlockwise rotation of 180 degrees as a result of multiplying by -1 , or, indeed, to a clockwise rotation of 180 degrees as a result of dividing by -1 (take your pick!). Thus, following the examples just given, the two square roots of 3 + 4i , i.e. 2 + i and -2 - i , are located 180 degrees apart, on opposite sides of the origin; similarly for the two square roots of 3 - 4i , i.e. 2 - i and -2 + i .

SQUARE ROOT OF A QUATERNION

Following the same basic idea as for finding the square root of a complex number (above), let's now attempt to produce a formula for the square root of the general quaternion, w + xi + yj + zk , where w, x, y, and z are all real numbers.

Suppose that p + qi + rj + sk is such a square root (with p, q, r, and s all real numbers); then

(p + qi + rj + sk)² = w + xi + yj + zk

We have no choice but to multiply the left-hand side out. (Sorry!)

p² + pqi + prj + psk + pqi - q² + qrk - qsj + prj - qrk - r² + rsi + psk + qsj - rsi - s²

= p² - q² - r² - s² + 2pqi + 2prj + 2psk

Well, it could be worse. So:

p² - q² - r² - s² + 2pqi + 2prj + 2psk = w + xi + yj + zk

Equating coefficients:

p² - q² - r² - s² = w         2pq = x         2pr = y         2ps = z

The last three of these give

q = x/2p         r = y/2q         s = z/2p

which we can substitute into the first one:

p² - x²/4p² - y²/4p² - z²/4p² = w ,    therefore     4p⁴ - (x² + y² + z²) = 4wp²

or     4p⁴ - 4wp² - (x² + y² + z²) = 0

(Note the similarity of this biquadratic equation to the corresponding equation in the complex case, above.)

Treating this as a quadratic equation in p² , and applying the quadratic formula:

p² = {4w ± sqrt[16w² + 16(x² + y² + z²)]} / 8

= {w ± sqrt[w² + x² + y² + z²]} / 2

Now, since we are insisting that p be real, and since {w - sqrt[w² + x² + y² + z²]} must be negative (unless x, y, and z are all 0, in which case the value of this expression will also be 0), we can simply drop the "-" from the "±". Then, taking square roots:

p = ±sqrt{w + sqrt[w² + x² + y² + z²]} / sqrt(2)

To make life a bit easier, let's refer to sqrt{w + sqrt[w² + x² + y² + z²]} as N, to save having to keep spelling it out. Then:

p = ±N / sqrt(2)

Now substitute this into q = x/2p :

q = x / [2 × ±N / sqrt(2)] = ±x / [N × sqrt(2)]

Similar substitutions into  r = y/2p  and  s = z/2p  yield, respectively:

r = ±y / [N × sqrt(2)]    and     s = ±z / [N × sqrt(2)]

Thus, finally, the square roots of   w + xi + yj + zk  can be written as:

±[N + (x / N) i + (y / N) j + (z / N) k] / sqrt(2)

where N = sqrt{w + sqrt[w² + x² + y² + z²]}

WHEW! So, at last, we have our formula. Now, let's look at its implications.

Firstly, if one or more of x, y, and z is not equal to zero, then - whatever value w takes - N will be positive and the formula works fine. (If just one of x, y, and z is not equal to zero, then effectively we're just taking the square root of a complex number, and the formula essentially reduces to that for the complex case given above.)

Secondly, if all three of x, y, and z are equal to zero, then N = sqrt(2w) if w is positive [in which case the square root is just sqrt(w), as we'd expect]; and zero if w is zero or negative. Since the formula involves division by N, and we can't divide by zero, we need to look at these last two cases more carefully:

On the one hand, if w, x, y, and z are all equal to zero, then the whole quaternion is just zero, and so is its square root - without having to resort to using the formula at all.

On the other hand, if w is negative, and x, y, and z are all zero, things get a lot more interesting. Again, N is zero (can you see why?), so that, again, we can't divide by it. Thus we need to adopt another approach.

Suppose (p + qi + rj + sk)² = -M² , where p, q, r, s, and M (for modulus) are all real numbers (so that -M² is the negative real number w). Then:

p² - q² - r² - s² + 2pqi + 2prj + 2psk = -M²

Since we've set this up so that the right-hand side (i.e. -M²) is a real number, the left-hand side must also be a real number. This means that

2pqi + 2prj + 2psk = 0 ,  therefore  2p(qi + rj + sk) = 0

This means that either p = 0 , or all three of q, r, and s are equal to 0 .

If q = r = s = 0 , then p² = -M² . However, since p is real, this is impossible unless p is zero (as squares of non-zero real numbers are always positive, and -M² is negative unless M is zero.)

Hence p must be 0 . This means that

-q² - r² - s² = -M² ,  therefore   q² + r² + s² = M²

- which means that (q,r,s) is any point on the surface of a sphere of radius M, centred on the origin, in a three-dimensional space in which the three axes represent, respectively, imaginary quaternion parts.

(Recall this diagram from page 2.)

Thus we have now proved that, in quaternion terms, there are infinitely many square roots of any negative real number, which are all purely imaginary and equidistant from the origin; and, also, that negative real numbers are the only quaternions that have this property.

The amazing thing is that we can achieve this result by using little more than high-school level algebra (referred to by mathematicians as "elementary algebra")!

By the way: interestingly, the quaternion square root formula just presented doesn't have the problem mentioned earlier with respect to the corresponding formula for complex numbers. In the latter, it was necessary to multiply one term by the signum function to ensure a correct answer every time; but this isn't necessary with the former, in which the problem "takes care of itself". In that sense, somewhat surprisingly, the quaternion formula is more "robust" than its complex counterpart.

Recall that all this started with one man's attempt to generalize the rotational nature of complex number multiplication to three dimensions.

He certainly succeeded: each of the three imaginaries represents a rotational degree of freedom. The fact that it was necessary to go to four dimensions to achieve the result, and also the astonishing result about infinitely many square roots of negative real numbers, add an element of intrigue to the proceedings.

In addition, there's an irony about all this that appeals to me.

It's common to think, is it not, that our three-dimensional world is reality, and that dreamy ideas about a fourth dimension involve using our imagination. Yet the three "imaginary" quaternion dimensions correspond in a natural way to our "real" world, with the "real" quaternion dimension corresponding to the "imaginary" one!

Actually, it's this kind of consideration which lies at the heart of the "problem" with quaternions. When Gibbs and Heaviside simplified Maxwell's electromagnetism equations, they did so by effectively separating the real (scalar) components from the imaginary (vector) components. This led to the development of scalars and vectors as two quite distinct concepts.

I'll admit that, as a student, I always had trouble with the vector concept. I always had the feeling that there was something missing - that, somehow, we weren't being told the full story. Why were there two distinct kinds of product involving vectors (the scalar or "dot" product, and the vector or "cross" product), which were quite different and appeared to have nothing whatever to do with each other? Why was the dot product a scalar, and - even more confusing - why was the cross product a vector at right-angles to both of the vectors being multiplied; and furthermore: why, when the order of multiplication was reversed, did the resulting vector point in the opposite direction? Why should the universe have this asymmetry built into it?

I feel that an understanding of quaternion algebra goes at least part of the way to addressing these questions. It's almost as if the nature of the vector dot and cross products follows as a necessity: "How could it be any other way?".

FOR THE RECORD

Quaternions came before vectors. Indeed, it was Hamilton himself who originally coined the terms "scalar" and "vector", to describe the real and imaginary parts, respectively, of quaternions. It was only later that these concepts took on a life of their own, without reference to the quaternions that gave rise to them in the first place. I find it outrageous that students of the physical sciences are now usually taught only half of the full story; and I raise the issue of how many vested interests have been only too happy to maintain the deception; and again, I ask why...

(I'm not alone in my concerns. See this page to read an article by a kindred spirit. Also, click on the links and have a good look around the site - most interesting...)

WE NEED TO STAND UP FOR TRUTH AND HONESTY IN SCIENCE, COME WHAT MAY.

There's more, however. The quaternions are only one of several types of hypercomplex numbers. Have a look at this page in which some of these are listed, with links leading to more detail. Don't worry if you don't understand all the terminology (I don't understand a lot of it myself!); just have a look over the page to get an idea of the scale of the subject.

Two of these various types of algebras are the biquaternions and the octonions, which are both extensions of the quaternions - much as the quaternions are themselves an extension of the complex numbers, which are in turn an extension of the real numbers...

The biquaternions are somewhat like the quaternions; but instead of allowing only real coefficients, we now allow complex coefficients! This doubles the number of dimensions from four to eight.

The octonions are something else again! With this algebra (also eight-dimensional), we lose not only the Commutative Law of Multiplication (CLM), but also the Associative Law of Multiplication (ALM). This is a major alteration, having a far more profound impact than the loss of the CLM.

However, like the real and complex numbers and the quaternions, the octonions still form a normed division algebra - which basically means that, in spite of the extra "damage" inflicted by the loss of the ALM, a kind of division is still possible and makes some sort of reasonable sense. The octonions are the fourth - and last - such division algebra; even the biquaternions don't measure up in this regard.

These - and many even more peculiar structures - were investigated in the 19th century, after Hamilton started the ball rolling by introducing the quaternions. One of the main researchers was William Kingdon Clifford, after whom the "Clifford algebras" (which have even more dimensions!) are named.

The octonions were discovered by John T. Graves (to whom Hamilton wrote his famous letter - see above). To learn more, click on this link which leads to a paper by John Baez (a photograph of whom, with some friends by Dublin's Broome Bridge admiring the plaque to Hamilton, can be accessed via a link in page 2 of this article). Also, may I recommend a visit to John Baez's home-page, in which you will find plenty of links to thought-provoking stuff, not all of it mathematical. (In particular, may I recommend this article.)

UPDATE, Saturday, 21st April 2007

On this page:

http://www.math.oregonstate.edu/bridge/papers/

you will find links to a number of articles by Tevian Dray, a Professor of Mathematics (and one of those four visitors to the Broome Bridge), and his colleague Corinne A. Manogue, a Professor of Physics (both at Oregon State University), who have tackled the issue of how vectors in general - and vector calculus in particular - are often taught in a less-than-intuitive way, which makes it hard for students to grasp the concepts, as well as being less than totally helpful for scientists, who prefer a more obviously practical, "applied" approach than the traditional abstract, "pure" mathematical approach.

The penultimate one of these links is to a .pdf file entitled "The Geometry of the Dot and Cross Products", which addresses the issues I raised above. Well worth a look! I'm quite sure that had I seen something like this as a student, I'd have been able to latch on to the essential ideas much more quickly and easily.

(By the way: in addition to their shared interest in how mathematics is taught, and how it is used by physicists, Professors Dray and Manogue have a common research interest in the octonions and their possible applications.)

I'll just squeeze in one more link about the development of algebraic systems.

This excellent page gives a history of hypercomplex numbers from the seventh century up to the present, and covers in summary form just about everything I've said in these three pages about the development of the subject. It's easy to read, and gives a good, clear perspective. Highly recommended.

(I actually included that link about halfway down page 1 of this article, with reference to the origin of the mathematical use of the term "imaginary"; but now, having read page 2 and this page also, you may find that you can get a lot more out of it than previously.)

CONCLUSION

Well! This article on quaternions, like Topsy, "just growed" - to three pages.

I hope you've managed to get something out of it. As I'm sure you've realized by now, I hope that the whole world latches on to the very practical issues the subject raises, and sooner rather than later: an application of this theory to the production of cheap energy, consigning the oil industry to history as soon as possible - thus enabling a serious attempt to be made to scale back greenhouse gas production, and removing the perceived need for greedy rich nations to inflict their particular kind of barbarism on less powerful nations which just happen to have large oil deposits.

If abstract algebra can help make the heartfelt sentiment expressed by the slogan "NO BLOOD FOR OIL" actually come to pass, and also pull the rug out from under the resurgent nuclear power lobby at the same time, I'll be cheering. If you've caught my drift, and are inspired to help spread the word and draw attention to these matters, I'll be cheering even louder.

(If you don't have the foggiest notion what I'm talking about, please visit my Zero-point energy page to find out.)

Over to you!

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