Here the rule we have applied is (x, y) -> (y, -x). He mentioned the Clifford biquaternions ( split-biquaternions) as an instance of Clifford algebra.Once students understand the rules which they have to apply for rotation transformation, they can easily make rotation transformation of a figure.įor example, if we are going to make rotation transformation of the point (5, 3) about 90 ° (clock wise rotation), after transformation, the point would be (3, -5). Concerning general relativity, he expressed the Runge–Lenz vector. He cited five authors, beginning with Ludwik Silberstein, who used a potential function of one quaternion variable to express Maxwell's equations in a single differential equation. Next he used complex quaternions ( biquaternions) to represent the Lorentz group of special relativity, including the Thomas precession. He proceeded to kinematics of rigid body motion. Girard began by discussing group representations and by representing some space groups of crystallography. The essay shows how various physical covariance groups, namely SO(3), the Lorentz group, the general theory of relativity group, the Clifford algebra SU(2) and the conformal group, can easily be related to the quaternion group in modern algebra. Girard's 1984 essay The quaternion group and modern physics discusses some roles of quaternions in physics. Quaternions have received another boost from number theory because of their relationships with the quadratic forms. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions. For this reason, quaternions are used in computer graphics, computer vision, robotics, control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics, computer simulations, and orbital mechanics. In addition, unlike Euler angles, they are not susceptible to " gimbal lock". The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. However, quaternions have had a revival since the late 20th century, primarily due to their utility in describing spatial rotations. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to follow. A side-effect of this transition is that Hamilton's work is difficult to comprehend for many modern readers. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature on quaternions. There was even a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems.įrom the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. 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. At this time, quaternions were a mandatory examination topic in Dublin. The last and longest of his books, Elements of Quaternions, was 800 pages long it was edited by his son and published shortly after his death.Īfter Hamilton's death, the Scottish mathematical physicist Peter Tait became the chief exponent of quaternions. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted most of the remainder of his life to studying and teaching them. An electric circuit seemed to close, and a spark flashed forth. This letter was later published in a letter to the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Hamilton states:Īnd here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples. Graves, describing the train of thought that led to his discovery. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from Dunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery. Into the stone of Brougham Bridge as he paused on it. Left column shows premultiplier, top row shows post-multiplier. For other uses, see Quaternion (disambiguation). This article is about quaternions in mathematics.
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