Quaternion

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 as alive as applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.

Quaternions are generally represented in the form

where , in addition to are real numbers; in addition to i, j, and k are the basic quaternions.

Quaternions are used in pure mathematics, but also clear practical uses in applied mathematics, especially for calculations involving three-dimensional rotations, such(a) as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an option to them, depending on the application.

In advanced Clifford algebra division algebra to be discovered.

According to the Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The sedenions, the quotation of the octonions, relieve oneself zero divisors and so cannot be a normed division algebra.

The unit quaternions can be thought of as a option of a group array on the 3-sphere 3 that provides the multinational Spin3, which is isomorphic to SU2 and also to the universal cover of SO3.

History

Quaternions were presented by Hamilton in 1843. Important precursors to this make-up mentioned Euler's four-square identity 1748 and Olinde Rodrigues' parameterization of general rotations by four parameters 1840, but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was non published until 1900.

Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to put and subtract triples of numbers. However, for a long time, he had been stuck on the problem of multiplication and division. He could not figure out how to calculate the quotient of the coordinates of two points in space. In fact, Ferdinand Georg Frobenius later proved in 1877 that for a division algebra over the real numbers to be finite-dimensional and associative, it cannot be three-dimensional, and there are only three such division algebras: complex numbers and quaternions which have dimension 1, 2, and 4 respectively.

The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the belief behind quaternions were taking quality in his mind. When thedawned on him, Hamilton could not resist the urge to carve the formula for the quaternions,

into the stone of Brougham Bridge as he paused on it. 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.

On the coming after or as a calculation of. day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery. This letter was later published in a letter to the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science; Hamilton states:

And here there dawned on me the conviction that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted nearly of the remainder of his life to studying and teaching them. his son and published shortly after his death.

After Hamilton's death, the Scottish mathematical physicist Maxwell's equations, were talked entirely in terms of quaternions. There was even a professional research association, the Quaternion Society, devoted to the examine of quaternions and other hypercomplex number systems.

From 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. Vector analysis spoke the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature on quaternions. 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 Hamilton's work is unmanageable to comprehend for many innovative readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and unoriented to follow.

However, quaternions have had a revival since the slow 20th century, primarily due to their proceeds in describing spatial rotations. The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. In addition, unlike Euler angles, they are not susceptible to "gimbal lock". 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. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms.

P.R. Girard's 1984 essay The quaternion office and modern physics discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely SO3, the Lorentz group, the general theory of relativity group, the Clifford algebra SU2 and the conformal group, can easily be related to the Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions split-biquaternions as an spokesperson of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime. Among the fifty references, Girard included Alexander Macfarlane and his Bulletin of the Quaternion Society. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions.

The finding of 1924 that in rotation groups.