Hyperbolic Quaternions

Abstract

It is well known that quaternions are intimately connected with spherical trigonometry, and in fact they reduce that subject to a branch of algebra. The question is suggested whether there is not a system of quaternions complementary to that of Hamilton, which is capable of expressing trigonometry on the surface of the equilateral hyperboloids. The rules of vector-analysts are approximately complementary to those of quaternions. In this paper I propose to show how they can be made completely complementary, and that, when so rectified, they yield the hyperbolic counterpart of the spherical quaternions.

Hyperbolic Quaternion Formulation of Electromagnetism

“…we obtain the hyperbolic quaternions which was first suggested by Alexander MacFarlane in 1891 [62]. The products of basis elements of the hyperbolic quaternions satisfy the same relations for real quaternions given in eq.…”

Associative Space-Time Sedenions and Their Application in Relativistic Quantum Mechanics and Field Theory

“…Mathematically, these two bases are equivalent, and the different physical properties attributed to them are an important physical essence of our sedenionic hypothesis. In contrast to the previously discussed sedeonic algebra [20]- [23], which uses the multiplication rules of basic elements ′ n a and ′ n e proposed by A. Macfarlane [28], the multiplication rules for sedenionic basis elements a n and e n coincide with the rules for quaternion units introduced by W. R. Hamilton [29]. There is a close connection between these two basses.…”

Reformulation of Relativistic Quantum Mechanics Equations with Non-Commutative Sedeons

“…Apparently, such possibility of vector basis multiplication was pointed first by Macfarlane, A. [31]. Later the similar multiplication rules for matrix basis were applied by Paul W. 2] nd Dirac, P.A.M. [33] in their spinor equations.…”