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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 = ijk = −1

& cut it on a stone of this bridge.

In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations. This article describes the original invention and subsequent development of quaternions.

Hamilton's discovery

In 1843, Hamilton knew that the complex numbers could be viewed as points in a plane and that they could be added and multiplied together using certain geometric operations. Hamilton sought to find a way to do the same for points in space. Points in space can be represented by their coordinates, which are triples of numbers and have an obvious addition, but Hamilton had been stuck on defining the appropriate multiplication.

According to a letter Hamilton wrote later to his son Archibald:

Every morning in the early part of October 1843, on my coming down to breakfast, your brother William Edward and yourself used to ask me: "Well, Papa, can you multiply triples?" Whereto I was always obliged to reply, with a sad shake of the head, "No, I can only add and subtract them."

On October 16, 1843, Hamilton and his wife took a walk along the Royal Canal in Dublin. While they walked across Brougham Bridge (now Broom Bridge), a solution suddenly occurred to him. While he could not "multiply triples", he saw a way to do so for quadruples. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge:

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

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted the remainder of his life to studying and teaching them. He founded a school of "quaternionists" and popularized them in several books. The last and longest, Elements of Quaternions, had 800 pages and was published shortly after his death. See "classical Hamiltonian quaternions" for a summary of Hamilton's work.

Precursors

Hamilton's innovation consisted of expressing quaternions as an algebra. The formulae for the multiplication of quaternions are implicit in the four squares formula devised by Leonhard Euler in 1748; Olinde Rodrigues applied this formula to representing rotations in 1840.

After Hamilton

After Hamilton's death, his pupil Peter Tait, as well as Benjamin Peirce, continued advocating the use of quaternions. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described entirely in terms of quaternions. There was a professional research association which existed from 1899 to 1913, the Quaternion Society, exclusively devoted to the study of quaternions.

From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs and Oliver Heaviside. Both were inspired by the quaternions as used in Maxwell's A Treatise on Electricity and Magnetism, but — according to Gibbs — found that "… the idea of the quaternion was quite foreign to the subject." Vector analysis described the same phenomena as quaternions, so it borrowed ideas and terms liberally from the classical quaternion literature. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side effect of this transition is that works on classical Hamiltonian quaternions are difficult to comprehend for many modern readers because they use familiar terms from vector analysis in unfamiliar and fundamentally different ways.

Principal publications

• 1853 Lectures on Quaternions
• 1866 Elements of Quaternions
• 1873 Elementary Treatise by Peter Guthrie Tait
• 1878 Abbott Lawrence Lowell: Quadrics: Harvard dissertation:
• 1882 Tait and Philip Kelland: Introduction with Examples
• 1885 Arthur Buchheim: Biquaternions
• 1899 Charles Jasper Joly: Elements vol 1, vol 2 1901
• 1901 Vector Analysis by Willard Gibbs and Edwin Bidwell Wilson (quaternion ideas without quaternions)
• 1904 Cargill Gilston Knott: third edition of Kelland and Tait's textbook
• 1904 Bibliography prepared for the Quaternion Society by Alexander Macfarlane
• 1905 C.J. Joly's Manual
• 1940 Julian Coolidge in A History of Geometrical Methods, page 261, uses the coordinate-free methods of Hamilton's operators and cites A. L. Lawrence's work at Harvard. Coolidge uses these operators on dual quaternions to describe screw displacement in kinematics.

Octonions

Octonions were developed independently by Arthur Cayley in 1845 and John T. Graves, a friend of Hamilton's. Graves had interested Hamilton in algebra, and responded to his discovery of quaternions with "If with your alchemy you can make three pounds of gold [the three imaginary units], why should you stop there?"

Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843 presenting a kind of double quaternion that is nowadays often called an octonion, and showing that they were what we now call normed division algebraTemplate:Fact; Graves called them octaves. Hamilton needed a way to distinguish between two different types of double quaternions, the associative bi-quaternions and the octaves. He spoke about them to the Royal Irish Society and credited his friend Graves for the discovery of the second type of double quaternion. observed in reply that they were not associative, which may have been the invention of the concept. He also promised to get Graves' work published, but did little about it; Cayley, working independently of Graves, but inspired by Hamilton's publication of his own work, published on octonions in March 1845 – as an appendix to a paper on a different subject. Hamilton was stung into protesting Graves' priority in discovery, if not publication; nevertheless, octonions are known by the name Cayley gave them – or as Cayley numbers.

The major deduction from the existence of octonions was the eight squares theorem, which follows directly from the product rule from octonions, had also been previously discovered as a purely algebraic identity, by Ferdinand Degen in 1818.

Mathematical uses

Quaternions continued to be a well-studied mathematical structure in the twentieth century, as the third term in the Cayley–Dickson construction of hypercomplex number systems over the reals, followed by the octonions and the sedenions; they are also useful tool in number theory, particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the quaternion group, is also the simplest non-commutative Sylow group.

The study of integral quaternions began with Rudolf Lipschitz in 1886, whose system was later simplified by Leonard Eugene Dickson; but the modern system was published by Adolf Hurwitz in 1919. The difference between them consists of which quaternions are accounted integral: Lipschitz included only those quaternions with integral coordinates, but Hurwitz added those quaternions all four of whose coordinates are half-integers. Both systems are closed under subtraction and multiplication, and are therefore rings, but Lipschitz's system does not permit unique factorization, while Hurwitz's does.

Quaternions as rotations

Quaternions are a concise method of representing the automorphisms of three- and four-dimensional spaces. They have the technical advantage that unit quaternions form the simply connected cover of the space of three-dimensional rotations.

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. Tomb Raider (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth 3D rotation. Quaternions have received another boost from number theory because of their relation to quadratic forms.

Memorial

Since 1989, the Department of Mathematics of the National University of Ireland, Maynooth has organized a pilgrimage, where scientists (including physicists Murray Gell-Mann in 2002, Steven Weinberg in 2005, Frank Wilczek in 2007, and mathematician Andrew Wiles in 2003) take a walk from Dunsink Observatory to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.

References

• John C. Baez, "The Octonions", Bulletin of the American Mathematical Society, New Series, volume 39, number 2, pages 145–205. PDF link.
• John H. Conway, Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. AK Peters, 2003, ISBN 1-56881-134-9.
• G. H. Hardy and E. M. Wright, Introduction to Number Theory. Many editions.

Notes

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