History of quaternions

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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

Contents 1 The golden age 2 Turn of the century triumph of real Euclidean 3 space 3 Historical metaphysical 19th Century controversy 3.1 Sign of distance squared 4 Nature of space and time controversy 5 20th century extensions 6 References

[edit] The golden age

Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. According to the story Hamilton told, 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

suddenly occurred to him; Hamilton then promptly 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 proceeded to popularize 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, mostly based on the notation and vocabulary of Hamilton is different than what is used today.

[edit] 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 survive, with the weaker notation to become extinct.^{[citation needed]} 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 Gibb's 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.^{[citation needed]} 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^[1]), 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.

[edit] 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.

[edit] 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.^[2]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^[3] 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^[4] 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.^{[5][citation needed]}

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.^{[citation needed]}

[edit] 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.^{[citation needed]} 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.

[edit] 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 (Cliffor, Lie, tensors) include quaternions, so in that respect they've become quite "mainstream", so to speak.

[edit] References