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

$i^{2}=j^{2}=k^{2}=ijk=-1\,$

Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says:
Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for quaternion multiplication
i² = j² = k² = i j k = −1
& cut it on a stone of this bridge.

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 from what is used today.

'See main article:'Classical Hamiltonian Quaternions

Turn of the century triumph of real Euclidean 3 space

Unfortunately some of Hamilton's supporters, like Cargill Gilston Knott, vociferously 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.

The 19th century Darwinist mentality of the time, allowed the respective champion of Quaternion notation and modern vector notation to allow their pet notations to become embroiled in a battle to the death, with the intent that only the strongest notation would be 'fittest' and survive, with the weaker notation left to become extinct. Modern notation won the day.

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 that attempted to show that early modern vector notation which included dyadics could do everything that Hamilton's quaternions could. Gibbs was working too hard on statistical mechanics to help with the manuscript. His student, Wilson, based the book on his mentor's lectures.

The dyad product and the dyadics it generated also eventually fell out of favor their functionality being replaced by the matrix.

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.

Some early formulations of Maxwell's equations used a quaternion-based notation (Maxwell paired his formulation in 20 equations in 20 variables with a quaternion representation), but it proved unpopular compared to the vector-based notation of Heaviside. The various notations were, of course, computationally equivalent, the difference being a matter of aesthetics and convenience.

The 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 and installed into vector analysis. 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.

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 meditating on 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. 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 SCALER 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. Descartes called it an imaginary number. Hamilton objected to calling the square root of minus one an imaginary number. In Descartes day complex number was a polite term for imaginary number, but they meant the same thing.

When Hamilton speculated that there was not just one, but an infinite number of square roots of minus one, and took three of them to use as a bases for a model of three dimensional space, rivaling Cartesian Coordinates there immediately arose a controversy about the use of quaternions that escalated after Hamilton's death.

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. 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.

Comparison with modern vector notation

Around the turn of the 19th into the 20th century early text books on modern vector analysis did much to move standard notation away from that classical quaternion notation, in favor of modern vector notation based on real Euclidean three space.

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

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 from what 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. 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" 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, that quaternions multiplied into vectors had performed. A dyad was written in some early text books as AB 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, 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 i² = −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 Cℓ_3,0(R) with its defining Clifford property

e_αe_β = −e_βe_α for α 1≠ 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 19th century.

See also

• Quaternion Society (1899 - 1913)