William Kingdon Clifford was an incredibly eccentric mathematician, who was responsible for several advancements in mathematics. Ironically, his short paper On the Space Theory of Matter foreshadowed Einstein’s general theory of relativity by suggesting that energy and matter are manifestations of different curvatures of space. Had it not been for his death at the early age of 33, perhaps Clifford would have published further findings connected to this scientific enquiry. More famously, Clifford was responsible for furthering the works of Hamilton via the generalisation of quaternions, later assimilating this number system into his own geometric algebra, now known as Clifford algebra. Clifford’s contributions to mathematics were previously overlooked due to his aforementioned early death; however, it is critical to understand his fundamental role in the development of quaternions and how they contended with other geometric algebras of the time.
Quaternions were first proposed by the mathematician Hamilton in his 1853 book, Lectures on Quaternions. Hamilton’s aim was to extend the two-dimensional plane of complex numbers (z=a+bi) into three dimensions, whereby (a, b) becomes (a,b, c). The motivation was that, as complex numbers describe a plane, i.e. a two-dimensional space with coordinates (a, b), the new numbers should describe a three-dimensional space with coordinates (a,b, c). While this proved impossible, he discovered that an extension is possible to a four-dimensional ‘four-component complex number’, or a quaternion of numbers. The real part of the quaternion would be taken as the fourth dimension, while the imaginary part would be interpreted as a geometric three-dimensional space (three orthogonal imaginary dimensions). It was in 1843 when Hamilton understood that he needed one additional scalar value component for his algebra regarding complex vectors to work. Hamilton subsequently theorised a formula which defined how the complex unit vectors i, j and k related to both one another and their metric signature of −1: i2=j2=k2=ijk=−1.
His excitement upon this invention as he “felt the galvanic circuit of thought close” can still be seen at Brougham Bridge, Dublin, where Hamilton carved this equation. At around the same time, Maxwell’s theory of electromagnetism was becoming popular, encouraging Hamilton to think about his quaternions in geometric terms by investigating the connection between complex numbers and rotations in a plane. This geometric thought led to the introduction of the noncommutativity of quaternion multiplication, a concept later incorporated into Clifford’s algebra.
Clifford first came across quaternions in the late 1860s. His enrolment at Trinity College, Cambridge was what initially incited his interest in mathematics as a theoretical discipline. Consequentially, his expectations of becoming an “ardent High Churchman” were promptly undermined as he was exposed to Darwinism, debate, and sports, as well as a tendency to go above and beyond in his self-guided studies of mathematics. One of Clifford’s teachers, James Sylvester, commented on his lack of adherence to the curriculum offered by the university, stating that he could very well have been the strongest of his year “had he chosen to devote himself exclusively to the University curriculum instead of pursuing his studies … in a more extensive field.” Nevertheless, Clifford produced numerous Cambridge publications between 1863 and 1871. It was during this time that he found his passion for geometric thought, coined in his paper Analytical Metrics, which described the two theorems of geometry which he understood: one referencing only position, and the other referencing magnitude. His interest in geometry from his early publications was also apparent in his 1865 publication, On Triangular Symmetry, which developed ideas about the metrical relations of an equilateral triangle. Clifford studied geometric algebras in both Euclidean and non-Euclidean spaces, which enabled him to develop a generalisation of Hamilton’s quaternions.
Clifford’s Preliminary Sketch of Biquaternions paper, published in 1873, suggested a quaternion with four complex number components in contrast to the four real number components of Hamilton’s quaternions. His use of the word biquaternions holds the purpose of showing that a biquaternion q+ωr, where q and r are regular quaternions and ω as a non-real algebraic entity, with the property 𝜔2 = 0. The paper was divided into five sections, discussing mechanical systems, a generalisation of Hamilton’s algebra of quaternions, non-Euclidean geometries and geometrical scenarios where biquaternions are present. His motivation for creating biquaternions stemmed from the limitations of scalars and vectors when representing specific mechanical quantities and behaviours – there are many instances where positions are relevant to mechanical quantities, such as when a force acts on a rigid body along a fixed line of action. In this situation, Clifford proposed the term rotor, i.e. rotation and vector, and called for an algebra which may combine scalar, vector and rotor quantities. This extension of the idea of scalars and vectors was what enabled Clifford to define the idea of multivectors which, like quaternions, can unify scalars with other algebraic components. While Hamilton generalised the algebra of complex numbers to a four-dimensional quaternion algebra, it was Clifford who incorporated these as subalgebras alongside a Cartesian vector component. If Clifford algebra achieved such a ground-breaking means of describing motions in three-dimensional space, then why is it not commonly used in mathematics today?
The reason behind the obscurity of Clifford algebra lies mostly in the timeframe in which it was created. The late nineteenth century saw intense competition in developing algebras – sometimes referred to as the vector algebra war – resulting in Clifford’s work being diluted by a plethora of other vectorial systems. Clifford was merely a contender in a network of countless mathematicians who pushed their own vector formalisms into the field. With his early death, Clifford did not stand a chance against Gibbs’ three-vectors, for example, nor Minkowski’s four-vectors. During this time, mathematicians sought to find a vector system, initiated by Hamilton’s quaternionic vectors, and it is now clear after over a century that Clifford’s system is more suitable than that of Gibbs, which is now typically taught. This can be demonstrated, for example, by Clifford algebra’s ability to reduce Maxwell’s field equations into one equation in comparison to the four equations of the Gibbs vector system. Clifford’s system provides a much more simplified version of Maxwell’s electromagnetism without the addition of matrices, spinors and complex numbers which are necessary for Gibbs’ vector system. Another issue with Gibbs’ vectors is that it can only express scalars and vectors and his ‘cross product’ calculation does not work for four dimensions, nor is it preserved under reflection. Nonetheless, this system is useful in the sense that vectors can be added or subtracted in a spatial sense and create visual representations of direction and magnitude.
Clifford algebra and its affiliation to Hamilton’s quaternions provided an excellent description of three-dimensional space and, although it is not used in most mathematical curricula, many fields are beginning to use this vector system today, most interestingly for Fourier transforms and the newly emerging study of quantum computing. It is therefore evident that the development of quaternions by Hamilton, though not his most famous work, sparked the interest of a young William Clifford and inspired him to pursue geometric algebra in his race to win first place in the hunt for an effective vector system. And while he did not win, his elaboration upon quaternions will not be forgotten.
Written by Kat Jivkova
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