History of quaternions


History of quaternions

This article is an indepth story of the history of quaternions. It tells the story of who and when. To find out what quaternions are see quaternions and to learn about historical quaternion notation of the 19th century see classical quaternions

The golden age

Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. On October 16, he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

:i^2 = j^2 = k^2 = ijk = -1,

occurred to him; Hamilton carved this equation into the side of the nearby Brougham Bridge (now called Broom Bridge). This involved abandoning the commutative law, a radical step for the time. Vector algebra and matrices had yet to be developed.

Hamilton popularized quaternions with several books, the last of which, "Elements of Quaternions", had 800 pages and was published shortly after his death.

Reading works written before 1900 on the subject of Classical Hamiltonian quaternions is difficult for modern readers because the notation used by early writers on the subject of quaternions is different than what is used today.

'See main article:'Classical Hamiltonian Quaternions

Historical metaphysical 19th-century controversy

The controversy over quaternions was more than a controversy over the best notation. It was a controversy over the nature of space and time. It was a controversy over which of two systems best represented the true nature of space time.

Sign of distance squared

This controversy involves the question, how much is one unit of distance squared? Hamilton postulated that it was a negative unit of time.

In 1833, before Hamilton invented quaternions, he wrote an essay calling real number Algebra the Science of Pure Time. [ [http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=14&frames=0&view=50 Historical Math Monographs ] ] Classical Quaternions used what we today call imaginary numbers to represent distance and real numbers to represent time. To put it in classical quaternion terminology the square of every vector is a negative scalar [ [http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=221&frames=0&view=50 Historical Math Monographs ] ] In other words in the classical quaternion system a quantity of distance was a different kind of number from a quantity of time.

Descartes did not like the idea of minus one having a square root; he called it an imaginary [imaginary] number. Hamilton objected to calling the square root of minus one an imaginary number

Hamilton speculated that there was not just one square root of minus one, but an infinite number, and took three of them to use as a bases for a model of three dimensional space, rivaling Cartesian Coordinates.Fact|date=January 2008

Nature of space and time controversy

Hamilton also on a philosophical level believed space to be of a four dimensional or quaternion nature, with time being the fourth dimension. His quaternions importantly embodied this philosophy. On this last count of the 19 century debate Hamilton in the 21st century has been declared with winner. To an extent any model of space and time as a four dimensional entity on a metaphysical level, can be thought of as type of "quaternion" space, even if on a notational and computational level Hamilton's original four space has continued to evolve.

An element on the other side opposing Hamilton's camp in the 19th century debate believed that real Euclidean three space was the one and only true model mathematical model of the universe in which we live.Fact|date=January 2008 The 19th century advocates of Euclidean three space, have by the 20th century been proven wrong. Obviously in the 21st century the final chapter on the nature of space time has yet to be written. Hamilton was correct in suggesting that the Euclidean real 3-space, universally accepted at the time, might not be the one and only true model of space and time.

Turn of the century triumph of real Euclidean 3 space

Some of Hamilton's supporters, like Cargill Gilston Knott, opposed the growing fields of vector algebra and vector calculus (developed by Oliver Heaviside and Willard Gibbs, among others), maintaining that quaternions provided a superior notation.

Gibbs and Wilson's advocacy of Cartesian coordinates lead them to expropriate i, j, and k, along with the term vector first introduced by Hamilton into their own notational system. The new vector was different from the vector of a quaternion.

As the computational power of quaternions was incorporated into the real three dimensional space, the modern notation grew more powerful, and quaternions lost favor. While Peter Guthrie Tait was alive, quaternions had Tait and his school to develop and champion them, but with his death this trend reversed and other systems began to catch up and eventually surpass his quaternion idea. The book Vector Analysis written by Gibbs' student E. B. Wilson in 1901 was an important early work. It was the first text book length book on modern vector notation which included dyadics. Wilson demonstrated that modern vectors could do many of the same things that Hamilton's quaternions could.Fact|date=January 2008 Gibbs was working too hard on statistical mechanics to help with the manuscript. His student, Wilson, based the book on his mentor's lectures.

An example of the debate at the time over quadrantal versor appears in the quaternion section of the Wikipedia biography of the life and thinking of Arthur Cayley who was an avid early participant in these debates.

Maxwell's Equations 1865-1905, the speed of light, and Hamilton's death

The first formulations of Maxwell's equations used a quaternion-based notation (Maxwell paired his formulation in 20 equations in 20 variables with a quaternion representation [Maxwell 1873] ).

Heaviside and Gibbs developed a Euclidean R^3vector-based notation which incorporated some of Hamilton's work, but was based on three dimensional Euclidean space, with the idea of time being decoupled from the idea of space.

Hamilton died in 1865. That same year, it was shown that light was an electromagnetic wave that propagated according to the wave equation that Hamilton's new math was instrumental in developing.

In 1881 [Michelson, American Journal of Science,22, 1881 pg 120] and with far greater precision in 1887 [Miclelson and Morley, American Journal of Sciences,43,1887,p.333] Michelson and Morley did very precise measurements on the speed of light and determined that it moved at the same speed in all directions.

Two years later in 1889 [* G.F. Fitzgerald, The ether and the earth's atmosphere, Science 13, 390 (1889).] George Francis FitzGerald an Irish Professor at Trinity College the same school Hamilton had taught at, and well trained in quaternions easily predicted the solution that moving objects were foreshortened in the direction of travel. In the 20th century this theory contraction theory was experimentally verified again and again. Lorentz thankfully recognizes Fitzgerald's work and mentions him by name in his 1885 [Versuch einer Thoerie der elektrischen und optischen Erscheingungen in betweget Kopern Lieden, 1895 H A Lorentz] and again in 1904 [Electromagnetic Phenomena in systems moving with any velocity less that that of light, H A Lorentz, English version in porceedings of the Academy of Science of Amstrerdam, 6, 1904] papers.

In 1901, 35 years after Hamilton's death, the second edition of Elements of Quaternions, now a two volume set, appeared.

1905 - 1916 Quaternions in the Early Relativistic Era

In 1905, Einstein released his special theory of relativity. Einstein showed that FitzGerald had been correct, but rather than give up completely on the idea of Euclidean space, the special theory of relativity introduced a flat four dimensional space consisting of three real numbers and an imaginary time component. Quantum mechanics still uses this space.

It was obvious from the point of view of Quaternion Advocates lead in this period by McFarland that it was much easier to formulate special relativity in terms of quaternions. The Lorentz invariant was the scalar of the product of two quaternions. The gamma factor of the Lorentz transformations, was the square root of the scalar part of a quaternion squared. FitzGerald had most of the math in 1889.

1916 - 1933 General Relativity

In 1916, Einstein reached some of the same conclusions Hamilton had reached in the last century, and extended them. Einstein's original formulation of general relativity gives a metric tensor with a trace of minus two, corresponding to the same computation that Hamilton called the scalar part of the square of a quaternion. In 1916, Einstein released another paper, Hamiltonsches Princip und allgemeine Relativitatstheoriein Stizungsbericte der preussischen Akad. d Wissenshaften, 1916. Hamilton's principle refers to Hamilton's formulation of classical physics in a way that is alternative to Newton's methods, not to quaternions.

General Relativity in its original formulation however was based on Riemannian geometry, not quaternions. Conceptually Riemannian geometry is an extension of Euclidean geometry, but it replaces the idea of a flat three dimensional distance with that distance measured with a metric tensor. Riemannian space has the advantage that it can be extended to any number of dimensions. Riemann died in 1866, just one year after Hamilton so his ideas had been around for a while, by 1916.

Space and time in Riemann's view were much more symetrical, and represented as an array of four real numbers. In Riemannian space, time is a fourth spacial dimension, and Riemann imagined four dimensional objects, like hyper-cubes. Riemann's methods could therefore be extended to higher dimensions, which was viewed by its supporters as a great advantage, but this extensibility was at the expense of one very fundamental concept of quaternions, that space and time were fundamentally made up of exactly one dimension of time and three dimensions of space.

The quaternion was a scalar plus a vector, Riemann theory posited in its four dimensional incarnation an array of four real numbers. This was a subtle but very important difference that escaped many thinkers at the time. There was another problem, quantum mechanics was still based on the so-called pre-1916 conception of real space, and at the time Einstein hoped that he would find a grand unification theory that would unify the world of relativistic gravity with quantum mechanics, but Einstein died in 1955, never having found his unified field theory, and because of this unable to accept Quantum Mechanics.

Bertrand Russell in his 1925 book The ABC of Relativity, suggested that the real trouble was with the nature of time. "The supposed necessity of attributing gravitation to a force attracting the planets towards the sun has arisen from a determination to preserve Euclidean geometry at all costs. If we suppose that our space is Euclidean which in fact it is not, we shall have to call in physics to rectify the errors in our geometry [The ABC of Relativity, Bertrand Russell, Third Revised Edition, 1969, pg 124]

Russel was a very independent thinker, but he was influenced by the quaternionists' point of view. Russell speaks of the FitzGerald contraction, not the Lorentz contraction, and his diagrams showing how to calculate Lorentz transforms hark back to Hamilton's Elements of quaternions, rather than the hyperbolic approach taken by Minkowski. [The ABC of Relativity, Bertrand Russell, Third Revised Edition, 1969, pg 46]

1933 to 1945 the lights go out in Europe

In 1933, Hitler came to power in Germany, and the German language school of thinking was ripped apart. Many of Germany's greatest scientists left the country. Einstein moved to America in 1933.

1945 to 1973

After the war, the breach between the relativists and the particle physics camp, grew wider and wider. Particle physics made discover after discovery of sub atomic particle.

Einstein died in 1955. He never found a way to unify quantum mechanics and relativity. He came to believe that quantum mechanics was fundamentally wrong.

There was the still somewhat Euclidean Geometry realm of quantum mechanics, and then there was General Relativity. The two ideas had not been unified to everyone's satisfaction.

In 1973 came the "standard model", which divided the Universe into realms. Most experiments since then have agreed with the standard model and the pace of new discoveries slowed.

1973 to 2005

The price of computers drops. It is discovered that in computer applications, quaternions often are the most efficient method of transforming three and four dimensional space.

In 1980 Herbert Goldstein publishes the second edition of Classical Mechanics. He devotes two sections to something he calls a Q matrix. These are two by two matricies used for rotation in one chapter and for Lorentz transformations in another. The foot note on page 156 reads, "The connossiur of somewhat musty mathematics will recognize in Eq(4-73) a representation of Q as a matrix quaternion, a quantity invented by Sir Willian R. Hamilton in 1843. Here E zero is the (quaternion) scalar and the quantity in parenthesis is the vector of a quaternion".

Chapter 7-4 of Goldstein's book titled Further descriptions of the Lorentz transform then takes back up the subject of his Q matrix and demonstrates that it can preform Lorentz transformations by rotating in four dimensional space.

"Modern" vector analysis turns 100 years old in 2001.

2005 to present

Interest in classical ideas from the 19th century continues.

In his 2006 book, "the trouble with physics", Lee Smolin wrote: "There are several ideas that might help string theory its key problems but they have not been widely studied. One is the idea that the old number system called Octonians is the key to a deep understandingof the relationship between super symmetry and higher dimensions." [The trouble with physics, the rise of string theory, the fall of science, and what comes next. Lee Smolin,2006 pg 274]

In 2007 Trifonov reported having discovered that quaternions were an important solution to the Einstein Equations. [ [http://members.cox.net/vtrifonov/ Trifonov, Vladimir] (2007), "Natural Geometry of Nonzero Quaternions", International Journal of Theoretical Physics, 46 (2) 251–257, DOI: [http://dx.doi.org/10.1007/s10773-006-9234-9 10.1007/s10773-006-9234-9] ]

Comparison with modern vector notation

Around the turn of the 19th into the 20th century early text books on modern vector analysis [See Vector Analysis by Gibbs and E. B Wilson 1901] did much to move standard notation away from that classical quaternion notation, in favor of modern vector notation based on real Euclidean three space.

Hamilton's vocabulary and methods poached by Euclidean thinkers.

From the point of view of men like Tait and FitzGerald, classical vector of a quaternion along with its computational power was ripped out of the classical quaternion multiplied by the square root of minus one, then perverted [ [http://books.google.com/books?id=R5IKAAAAYAAJ&pg=PA337&dq=perversor Vector Analysis Gibbs pg.337] ] it and installed into Euclidean vector analysis. In the view of those arguing in favor of quaternions, Hamilton's 1840's thinking was far more modern than Euclid's 2,000 BC thinking.

The computational power of the tensor of a quaternion and the versor of a quaternion became the dyadic. The scalar and the three vector went their separate ways.

The 3 × 3 matrix the took over the functionality of the dyadic which also fell into obscurity.

The scalar-time, 3-vector-space, and matrix-transform had emerged from the quaternion and could now march forward as three different mathematical entities, taking with them the functionality of the 19th century classical quaternion. The old notation was left behind as a relic of the Victorian era.

Vector and matrix and modern tensor notation had nearly universally replaced Hamilton's quaternion notation in science and real Euclidean three space was the mathematical model of choice in engineering by the mid-20th century.

Reinterpretation of i,j,k

Cartesian Coordinates represented three space with an ordered triplet of real numbers, (x,y,z). Quaternion notation introduced a different representation for the vector part of a quaternion:


Vq = xi + yj + zk [ [http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05140001&seq=&view=50&frames=0&pagenum=64|A S Hardy Elements of Quaternions 1887] ]

Cartesian coordinates were separated by commas; but classical quaternions were separated by plus or minus signs, and considered the summation of numbers of different types.

Vector analysis appropriated this form, and indeed the expression above from 1887 looks a lot like a modern vector. But there was an important difference. In the quaternion system each of the terms i,j,k is a square root of minus one. But the vector system rejected this. In the vector system i,j and k remained to indicate different orthogonal unit basis vectors. But, unlike the quaternion basis, these were considered to be derived from real Cartesian coordinates, made up of only real numbers. The idea that in any sense i times i could equal minus one was rejected.

In vector notation, the i, j and k had come to mean something rather different than they had in quaternion notation.

Four new multiplications

The classical quaternion notational system had only one kind of multiplication. But in that system the product of a pure vector of the form 0 + xi +yj + kz with another pure vector produced a quaternion.

To add the functionality of classical quaternions to the real three space early modern vector analysis required four different kinds of multiplication. [See Vector Analysis by Gibbs and E. B Wilson 1901] In addition to regular multiplication which got the name "scalar multiplication" to distinguish it from the three new kinds, it required "two" different kinds of "vector products". The fourth product in the new system was called the "dyad product".

Dot product

The first new product was called the "scalar product" [Vector Analysis Gibbs-Wilson 1901 pg 55] of two vectors, and was represented with a dot. Computationally it was equivalent to the operation of taking the negative of the scalar part of the quaternion product of the vector parts of two quaternions, so the new a · b corresponded to the old quaternion operation −S(VA × VB).

This meant that i · i in the new system was +1. And the type of "vector" in the modern system was different as well; the new "vector" was not the vector of the classical quaternion system, because it did not consist of a triplet of imaginary components. Rather it was a "modern vector" which had been striped of the classical property that the product was ii = -1.

Cross product

The second new product in the new system was the cross product, that was computationally similar to V(VA × VB) or taking the vector part of the product of the vector part of two classical quaternions.

In the new notational system it was still true that (i × j)= k, however, unlike (i · i)=+1 in the second new type of multiplication (i × i)=0.

Dyadic product

The third new product, in the early modern system was the dyad product. It was needed to perform some of the linear vector functions, [See Vector Analysis by Gibbs and E. B Wilson 1901] that quaternions multiplied into vectors had performed. A dyad was written in some early text books as AB [See Vector Analysis by Gibbs and E. B Wilson 1901] without a dot or cross in the middle. Three dyads made up a dyadic. This vector product took over the quaternion operations of version and tension. This early aspect of the Gibbs/Wilson system has become more obscure over time.

New system questioned

In the 19th century supporters of classical quaternion notation and modern vector notation debated over which was best notational system, as described above.

To provide a vastly oversimplified, short introduction to what motivated these debates consider that in the new notation that i · i =+1, j · j =+1 and k · k =+1. So apparently i,j,k in the modern vector notational system represent three new square roots of positive one.

In the new notational system i, j, and k also apparently represented square roots of zero, since i × i = 0 , j × j = 0, k × k = 0. The new notation system was then based on numbers that were the square root of both zero and positive one. Advocates of the classical quaternion system liked the older idea of a single vector product with a unit vector multiplied by itself being negative one better.

The quadrantal versor argument

An important argument in favor of classical quaternion notation was that i, j, and k doubled as quadrantal versors. i × (i × j) = −j and (i × i) × j = −j This was not the case in the new notational system of modern vector analysis because their cross product was not associative. In the new notation (i × i) × j = 0, and however i × (i × j) = −j.

Turn of the century triumph of modern vector notation

Modern vector notation eventually replaced the classical concept of the vector of a quaternion.

Advocates of Cartesian coordinates expropriated i, j, and k, along with the term vector into the modern notational system. The new modern vector was different from the vector of a quaternion.

As the computational power of quaternions was incorporated into modern vector notation, [Vector Analysis] classical quaternion notation lost favor.

The classical vector of a quaternion was multiplied by the square root of minus one and then again by negative one, and installed into modern vector analysis. The computational power of the classical quaternion vector product was exported into the new notation as the new cross and dot products. The computational power of the tensor of a quaternion and the versor of a quaternion became the dyadic, and then the matrix. The scalar and the three vector went their separate ways.

The 3 × 3 matrix rotation matrix took over the functionality of the dyadic which also fell into obscurity.

The scalar-time, 3-vector-space, and rotation matrix-transform had emerged from the classical quaternion and could now march forward as three different mathematical entities, taking with them the functionality of the 19th century classical quaternion. The old notation was left behind as a relic of the Victorian era.

Modern vector and matrix and modern tensor notation had nearly universally replaced Hamilton's quaternion notation in science and real Euclidean three space was the mathematical model of choice in engineering by the mid-20th century.

20th-century extensions

In the early 20th century, there has been considerable effort with quaternions and other hypercomplex numbers, due to their apparent relation with space-time geometry. Hypercomplex number, coquaternion, or hyperbolic quaternion, just to mention a few concepts that were looked at.

Descriptions of physics using quaternions turned out to either not work, or to not yield "new" physics (i.e. one might just as well continue to not use quaternions).

The conclusion is that if quaternions are not required, they are a "nice-to-have", a mathematical curiosity, at least from the viewpoint of physics.

The historical development went to Clifford algebra for multi-dimensional analysis, tensor algebra for description of gravity, and Lie algebra for describing internal (non-spacetime) symmetries. All three approaches (Clifford, Lie, tensors) include quaternions, so in that respect they've become quite "mainstream", so to speak.

Modern synthesis

Quaternions have had a revival in the late 20th century, primarily due to their utility in describing spatial rotations. Representations of rotations by quaternions are more compact and faster to compute than representations by matrices. For this reason, quaternions are used in computer graphics, control theory, signal processing, attitude control, physics, bioinformatics, and orbital mechanics. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions.

An important conceptual step forward has been the wider realisation that the "vector part" of quaternions most naturally represents not vectors in 3D but pseudovectors, the Hodge duals of vectors. Pseudovectors in 3D (also known as bivectors) are associated with the direction of oriented 2D planes. A difference with vectors is that whereas vectors change sign under co-ordinate inversion, a → −a, b → −b, pseudovectors in 3D (bivectors) remain unchanged, a ^ b → −a ^ −b = a ^ b.

The two systems, vector and quaternion, are combined by grafting a new element onto the quaternion system, the unit pseudoscalar "i", that has the property "i"2 = -1, and is defined to multiplicatively commute with the quaternions i, j and k.

This generates three further new linearly independent bases for the algebra, conventionally e1 = − "i" i, e2 = − "i" j and e3 = − "i" k, which have the property that :(e1)2 = (e2)2 = (e3)2 = +1, and linear combinations of them a and b have the product :ab = a · b (scalar) + "i" a × b (pseudovector). The new bases therefore behave appropriately for the basis elements of a space of modern vectors, with the unit pseudoscalar "i" identifiable as the unit scalar triple product (e1 ^ e2 ^ e3).

This augmented system, which is in fact a geometrical interpretation of the Clifford algebra "C"ℓ3,0(R) with its defining Clifford property :eαeβ = −eβeα for α≠β,restores to the vectors the single product of Hamilton; and preserves the structure of the Hamiltonian quaternions with all their rotational magic; but it also cleanly distinguishes the different types of geometric objects – scalars, vectors, pseudovectors, and pseudoscalars; and satisfies the wish of the vector pioneers for vectors which square to +1, rather than -1. Best of all, it generalises readily to any number of dimensions of the underlying ground space (taking quaternion-like rotation techniques with it), and so quaternions are no longer left perceived as isolated, a strange freak of the 3D world.

This re-integration into the mainstream has led to a renewed rediscovery and interest in the techniques for geometry pioneered by the classical Hamiltonian quaternion methods in the nineteenth century.

References


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