Professor Colleen Halverson
17 May 2013
The Quaternion Debate
Quaternion, just the word sounds strange. It is not without difficulty that it rolls of ones tongue for the first time. It almost conjures up the image of a race of evil, small, green aliens bent on galactic domination. Actually, quaternions are nothing that sinister. The word is a name for a somewhat obscure form of mathematics. The obscurity of quaternion algebra is due to the hazy results of a rather unorganized, polarized debate over their use and meaning that took place in the letters to the editor of the magazine, Nature, at the end of the nineteenth century. In fact, this debate still exists today, the platform having expanded to encompass areas of computer programming, weather science, quantum physics and new theories on gravity.
Quaternion algebra is a system of math describing spatial geometry. A quaternion in the singular sense is set of four numbers representing values in a given direction. The letters i j, and k are used to denote the same directions as x, y, and z in the 3-dimensional coordinate system. In the quaternion represented by q = a + bi + cj + dk there is a quantity with a magnitude value of a, that is b units in the i direction, c units in the j direction and d units in the k direction. Here i, j and k exhibit mathematical similarities to what is known as the “imaginary number”.
The square root of negative one is called the imaginary number because in mathematics, with what are consequently called the real numbers, the square of a number is always positive. To get a square root of a negative number, mathematicians came up with i, where i2 (i multiplied by i ) is negative one. In quaternion algebra, j and k are similar to i.1 Hamilton’s formula that governs the mathematics of quaternions follows the following rules: i2=j2=k2=ijk=-1. Quaternions are also noncommutative. With real numbers, two times four is the same as four times two. This is what is known as the commutative property. However, with quaternion arithmetic, a different multiplication order will result in a different answer. For example, ij = -ji (Hamilton 2).
An Irish mathematician by the name of Sir William Rowan Hamilton first discovered quaternions (McDonald 1). In the mid-1800s, Hamilton was trying to understand how to multiply numerical components, called “ordered triples”, or points in 3-dimensional space. Thomas L. Hankins, author of the biography, Sir William Rowan Hamilton, says that, “Hamilton’s ‘triadic fancies’, as he called them, were in part a reflection of his growing interest in metaphysics and in part a reflection of his frustration at being unable to extend his theory of number couples to three dimensions” (291). The story goes that, while he was working to uncover the quaternions, he would come down stairs every morning to be greeted by his elder son. “Well Papa, can you multiply triplets?” to which Sir William would regrettably confess no; he could only add and subtract them (Hankins 291). One morning in the October of 1843 while walking with his wife across a bridge near Dublin, Ireland, the answer hit him in a flash. He was so inspired by the sudden insight that he said he could not “resist the impulse-unphilosophical as it may have been-to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula” (Hankins 293).
James Clerk Maxwell, largely regarded as the father of electricity and magnetism, helped spread the news about quaternions by using them in his Treatise on Electricity and Magnetism published in 1873. Maxwell had been introduced to quaternions by Hamilton’s chief disciple, Peter Guthrie Tait. After reading Maxwell’s treatise, both Josiah Willard Gibbs from Yale and Oliver Heaviside in England became inspired to develop their own system of mathematics as, what they saw was, a simpler alternative to using quaternions. Their alternative, “vector analysis”, as it became know, is now integral to modern physics (Hankins 316).
Since their birth, quaternions have been followed by controversy. On the one hand there are those that use quaternions and are quick to extol their significance and the genius of Hamilton. On the other there are those that almost scoff at their absurdity and deem them pointlessness and difficult to use. The majority of the public debate concerning the use of quaternions took place through letters to the editor in the magazine, “Nature”, between the years 1880-1900. The leaders of this debate were Heaviside and Gibbs, defending their newly developed vector analysis, and Tait defending his mentor’s algebra of quaternions. When one looks at how little people have ever heard the word “quaternion”, it would seem that the “quaternionists” lost this debate. However, how this debate was won is not immediately clear.
Operations with quaternions can be difficult both mechanically and conceptually, although this was not true for their discoverer, Hamilton. “He could manage the full panoply of quaternion operations with ease: and while lesser minds quailed before their complexity, he felt no reason to sacrifice their beautiful algebraic properties for what P. G. Tait called the ‘hermaphrodite monster’ of vector analysis” (Hankins 316). In reading the letters to the editor in Nature during the time this debate took place, finding such strong phrases as “hermaphrodite monster,” is not uncommon.
“By 1893 the battle between the quaternionists and the vector analysts was in full swing. It was really two battles, one of quaternions versus coordinates, and a second one of quaternions versus vectors… In 1890 Tait entered the controversy on both fronts” (Hankins 319). The dialogue that took place through the Nature letters reveal an impassioned debate, that at times, almost degrades to ridicule in a back and forth exchange following from letter to letter. In one instance, Heaviside remarks at the pleasure he “derived from Prof. Gibbs attacks upon the quaternionic system in the rather one-sided discussion that took place…” in a previous edition of Nature (533). He also seems somewhat happy in reporting, “There is confusion in the quaternionic citadel; alarms and excursions, and hurling of stones and pouring of boiling water upon the invading host” (533-34). This last quote was a direct attack on P. G. Tait, whom, Heaviside is quick to point out, was “unable to bring his massive intellect to understand my vectors or Gibbs’s…” (534). Here it seems that the essence of what Heaviside is saying gets lost in his sarcasm.
Hankins points out in Hamilton’s biography that indeed, Tait may not have possessed a complete understanding of quaternions. He says, “Hamilton would not have agreed with all aspects of Tait’s defense. In particular, Tait criticized a statement by Gibbs to the effect that quaternions were limited to a representation of three-dimensional space “ (321). In the letter that Hankins makes reference to above, Tait asks, “What have students of physics, as such, to do with space of more than three dimensions?” (“The Role of Quaternions” 608). Tait made this statement in 1891, twenty-six years after Hamilton died. Fourteen years later, Albert Einstein published his famous paper on the special theory of relativity. In 1907, “the mathematician Herman Minkowski reformulated Einstein’s relativity paper into a theory of space-time, a four-dimensional space” (Baierlein 199-201). More recently, physicists have been "working on string theory, which views everything in the universe as harmonic vibrations of strings in eleven dimensional space…” (Cole 12). In a paper published in 2011, Jochem Hauser and Walter Dröscher examine the possibilities of gravity powerd propulsion systems based on recent experimental evidence and an emerging theory known as Poly Metric Tensor theory, or PMT theory. In PMT theory the existence of new particles of matter are predicted with the use of quaternions in an eight-dimensional, “internal space” (293-96). It looks like physicists have plenty to do in more than three dimensions.
After reading more about Tait, one is left with the image of a young apprentice, not quite mastered in the art, who feels he must carry on his dead master’s legacy. Hankins writes, “As the most direct disciple of Hamilton he may have felt responsible for preserving the ‘quaternion stream pure and undefiled’…” (321). Perhaps, having caught a glimpse of the “beautiful algebraic properties” of quaternions, Tait felt an obligation to garrison the “quaternionic citadel” and preserve the mathematic beauty of quaternions for future generations. Given the apparent aggressiveness of Heaviside’s assault, and the fact that Tait was quite outnumbered, it’s not hard to see how his confidence could have been shaken, causing him to falter in his defense of his mentor’s discovery.
While the path of dialogue between Tait and Heaviside reveals hostility and sarcasm, the dialogue between Tait and Gibbs appears much more civilized. This is in part, no doubt, due to Gibbs sticking with the facts and not resorting to sarcastic insults. In a response to a paper published by one of Tait’s few supporters, Prof. C. G. Knott, Gibbs first states the charges brought against his position by his critic, and then, in calm and clear language defends himself against the charges. In discussing the finer differences between vector analysis and quaternions, he opens up the possibility of middle ground between him and Knott. Gibbs writes, “Perhaps Prof. Knott will say that since I use both of them it matters little whether I combine them or not. If so I heartily agree with him” (364). This attitude of Gibbs seems to have had a reciprocal effect in Tait. In commenting on the dialogue between him and Gibbs, Tait writes, “My remark about Prof. Willard Gibbs was meant in all courtesy, and I am happy to find it so taken by him. The question between us, being thus a scientific one only, can afford to wait for a fortnight or so :- until my present examination season is past (“Quaternions” 535). Here Tait hits upon a deep wisdom, addressing scientific debate, as it should be, with calmness and courtesy. Hankins hits it right on the nose when he writes, “For the most part the battle of vectors versus quaternions was an argument with out a point” (321).
The most common place to find quaternions today is in computer programming. Yet, even today with computers, controversy still follows the quaternions. In an article written for the website gamedev.net, Diana Gruber, a senior programmer for Ted Gruber Software, Inc. explains why it is that programmers don’t need quaternions. The article was prefaced with the editor saying,“This has been, without question, the most controversial article we've ever posted…” and at the very bottom of the page under the comments section was posted, “Note: Please offer only positive, constructive comments - we are looking to promote a positive atmosphere where collaboration is valued above all else” (Gruber). The article is written in mathematical language and out lines the ways in which quaternions and vectors do essentially the same thing. She also goes into some of the same history discussed here. However, at the end of the article, Mrs. Gruber created a FAQ (frequently asked questions) section where a familiar tone of sarcasm comes across. The section is prefaced with, “Diana is tired of responding to the same questions and comments over and over and so has prepared this canned response. If your question/comment about quaternions is not answered here, then don't worry about it. It probably isn't very important anyway”. Her last two entries in the FAQ read thus:
“8. Contact that Shoemake guy, that Eberly guy, that _______ guy. He will straighten you out.
A. No thank you.
9. You have done a grave disservice to mathematics/physics/the game development community/newbie programmers/your company/your reputation.
A. I think they will all survive” (Gruber).
There seems to be a similar sentiment of sarcasm coming across here that Heaviside presented in the Nature letters. The author fails to see how 8 & 9 of Gruber’s FAQ are important questions.
Despite the fact that debate over their usefulness still rages on, quaternions have found their way into the framework of some modern theories, particularly concerning gravity. T. N. Palmer, a Royale Society Professor of Physics at the University of Oxford uses quaternions in what he calls The Invariant Set Postulate. In this postulate, Palmer brings together knowledge from weather models and chaos theory and joins that with fractal geometry and quaternions to give an explanation for some of the most difficult conceptual problems in quantum physics (Palmer, “Quantum Reality” 530). Palmer thinks that this postulate “…provides a possible realistic perspective on the essentials of complex numbers and quaternions in quantum theory” (“Invariant Set” 3165). The world is a lot bigger than it was in the late 1800’s and one would be hard pressed to find a similar setting as the Nature letters to stage a similar debate in this day and age, but nevertheless, the corridors of history seem to be sounding a faint echo. Quaternions; they are either essential, or they are useless, and there does not seem to be any middle ground.
So, what is it about quaternions that drive people so far one way or the other? Why, after over one hundred years later does this controversy remain? Perhaps the image of tiny, green, menacing aliens is too strong for some. Most likely, it is the way quaternions incorporate the “imaginary number”, the square root of -1. Heaviside alludes to this when he makes a pun in discussing his frustrations with “…the inscrutable negativity of the square of a vector in Quaternions; here, again, is the root of the evil”(534). There is a grave difficulty for some in taking seriously some thing that is called imaginary. For physicists and mathematicians like Heaviside, imaginary numbers are for those wishing to enjoy playtime in the dollhouse of the “quaternionic citadel”.
The number i has been “used by mathematicians for hundreds of years and was adopted… in the early 1800s for use in non-Euclidean or non-flat geometry” (Seifer 70). When Minkowski reformulated Einstein’s relativity paper to incorporate the concept of a fourth-dimensional space-time, he did so by using the number i. “By introducing a time-dimension equivalent to the three space coordinates, a ‘happening in 3-d space physics becomes, as it were, an existence in the 4-dimensional world’” (Seifer 69).
One of the consequences that follow from the mathematical framework of relativity is that nothing can travel faster than the speed of light. During this same time, physicists were trying to understand what was going on with electrons inside the atom. The current theory at the time stated ”…that in order to produce the necessary magnetic field, the electron would have to rotate so fast that the points on its equator would be spinning at much higher velocities than the speed of light!”(Seifer 71). Paul Dirac, an English theoretical physicist, pondered this problem and “… created an elegant solution for the spinning electron which involved the imaginary unit i, [and] the problem of violating relativity was sidestepped” (Seifer 72-73).
In his book Transcending the Speed of Light, Mark Seifer argues that science needs to develop a paradigm that incorporates the phenomena of consciousness. To Seifer, “the use of imaginary numbers to explain physical phenomena, such as by Minkowski and Dirac, makes sense, and it supports the notion that mind cannot be separated from matter. Thus, irrational and imaginary numbers are needed to explain higher-order descriptions of physical reality” (Seifer 143). In his 1921 letter to the editor in Nature, E. H. Synge, an Irish mathematician contemporary to Dirac and Minkowsky, writes, “It is, perhaps not generally realized that the theory of space and time, to which Minkowski was led on experimental grounds, had been formulated on general principles sixty-five years previously by Hamilton, the Irish mathematician… It is curious, therefore, that there should be a lack of recognition that the world of Minkowski is in all points identical with the system of quaternions of Hamilton…” (Synge 693). It is interesting to note that at the beginning of this paper, Hankins was quoted saying Hamilton’s renewed efforts that he put forth in the October of 1843, were in part a reflection of his growing interest in metaphysics. Perhaps Hamilton had an inkling of where his quaternions would lead to in the future.
Although the debate that took place in the letters to the editor in Nature in the late 1800s may not have been carried out in the most civilized way, the vector analysis of Gibbs and Heaviside is what physicists use today. Perhaps if the quaternion guru, Hamilton, had been alive, the debate would have turned out differently. Given that vector analysis has been used for so long, and that the study of physics stands at the forefront of evolving technology, it seems quite clear that vector analysis work very well. With this insight, it seems perfectly rational to side with Gibbs and Heaviside and stay with what works. There is an old saying, “If it’s not broke, don’t fix it!” That certainly could be applied in the case of quaternions. However, physicist’s understanding of reality is far from complete.
In the beginning of Maxwell’s book, The Dynamical Theory of the Electromagnetic Field, Einstein wrote a small section in appreciation of Maxwell for a republication of the book after Maxwell’s death. Einstein starts out by saying, “The belief in an external world independent of the observing subject lies at the foundation of all natural science. However, since sense-perceptions only inform us about this external world, or physical reality, indirectly, it is only in a speculative way that it can be grasped by us. Consequently our conceptions of physical reality can never be final. We must always be ready to change these conceptions, i.e. the axiomatic basis of physics, in order to do justice to the facts of observation in the most complete way that is logically possible” (29). Any scientist in any field would do well to take these words to heart.
In The Structure of Scientific Revolutions, Thomas Kuhn concludes with this insight into the nature of the scientific process: “At the start a new candidate for a paradigm may have few supporters, and on occasions the supporters’ motives may be suspect. Nevertheless, if they are competent, they will improve it, explore its possibilities, and show what it would be like to belong to the community guided by it. And as that goes on, if the paradigm is one destined to win its fight, the number and strength of the persuasive arguments in its favor will increase. More scientists will then be converted, and the exploration of the new paradigm will go on. Gradually the number of experiments, instruments, articles, and books based upon the paradigm will multiply” (159).
Quaternions represent such a paradigm, though they are hardly new. Its seems that they have been rather slow in passing through Kuhn’s stage of gathering supporter. However few, there are still supporters, and despite fierce debate, they seem confident enough to continue on their current path of inquiry. Perhaps the future will be even more unkind to Hamilton’s quaternions and they will wither away from the collective human memory forever. On the other hand maybe they are just the ticket to trigger a much needed revolution in scientific understanding. Only time will tell. However, if there is one thing science can learn from the quaternion debate, it is a reminder to stay objective. Should quaternions really prove to be useless, than they will become obsolete naturally. If, on the other hand, they are still being used, then there is no need to throw them out the window in ridicule just yet. The author feels that the editor who posted the comment on the bottom of the webpage containing Gruber’s article put it just right: “Note: Please offer only positive, constructive comments - we are looking to promote a positive atmosphere where collaboration is valued above all else”. This statement is in the true spirit of scientific inquiry.
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