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Quaternions, as an algebra, were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Hamilton knew that the complex numbers could be viewed as points in a plane, and he was looking for 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 for many years Hamilton had known how to add and multiply triples of numbers. But he had been stuck on the problem of division: He did not know how to take the quotient of two points in space.

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. He could not divide triples, but he could divide 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:

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

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.

After Hamilton's death, his pupil Peter Tait continued promoting quaternions. At this time, quaternions were a mandatory examination topic in Dublin, and in some American universities they were the only advanced mathematics topic taught. 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 even a professional research association, the Quaternion Society (1899 - 1913), exclusively devoted to the study of quaternions.

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 Leonard Euler in 1748; Olinde Rodriques applied this formula to representing rotations in 1840.

Vector analysis

From the mid 1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs and Oliver Heaviside. 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.

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: Lipshitz 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. Quaternions have received another boost from number theory because of their relation to quadratic forms.

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, 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 H. Conway, Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. AK Peters, 2003, ISBN 1568811349.
• G. H. Hardy and E. M. Wright, Introduction to Number Theory. Many editions.

Notes

1. John H. Conway, Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. AK Peters, 2003, ISBN 1568811349, p. 9
2. Hardy and Wright, Introduction to Number Theory, §20.6-10n (pp. 315-6, 1968 ed.)
3. John H. Conway, Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. AK Peters, 2003, ISBN 1568811349, chapter 2.
4. Ken Shoemake (1985), Animating Rotation with Quaternion Curves, Computer Graphics, 19(3), 245-254. Presented at SIGGRAPH '85.
   Tomb Raider (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth 3D rotation. See eg Nick Bobick, "Rotating Objects Using Quaternions", Game Developer magazine, February 1998