Related papers: Algebra and geometry of Hamilton's quaternions
Quaternions, discovered by Sir William Rowan Hamilton in the 19th century, are a significant extension of complex numbers and a profound tool for understanding three-dimensional rotations. This work explores the quaternion's history,…
Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point…
To construct ternary "quaternions" following Hamilton we must introduce two "imaginary "units, $q_1$ and $q_2$ with propeties $q_1^n=1$ and $q_2^m=1$. The general is enough difficult, and we consider the $m=n=3$. This case gives us the…
Quaternion, an extension of complex number, is the first discovered non-commutative division algebra by William Rowan Hamilton in 1843. In this article, we review the recent progress on building up the connection between the mathematical…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
In this paper I will give a brief history of the discovery (Hamilton, 1843) of quaternions. I will address the issue of why a theory of triplets (the original goal of Hamilton) could not be developed. Finally, I will discuss briefly the…
Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius' theorem, which says that "the only finite-dimensional real division…
In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions that closely resembles Minkowski spacetime. Viewing this in a modern context, we expand upon Macfarlane's idea and develop a model for real hyperbolic 3-space in…
The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real…
Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the…
Rotations on the 3-dimensional Euclidean vector-space can be represented by real quaternions, as was shown by Hamilton. Introducing complex quaternions allows us to extend the result to elliptic and hyperbolic rotations on the Minkowski…
In the past, Kepler painstakingly derived laws of planetary motion using difficult to understand and hard to follow techniques. In 1843 William Hamilton created and described the quaternions, which extend the complex numbers and can easily…
150 years after the discovery of quaternions, Hamilton's conjecture that quaternions are a fundamental language for physics is reevaluated and shown to be essentially correct, provided one admits complex numbers in both classical and…
Is it possible to define, for certain values n the product of vectors of the real vector space of n dimensions, such that this is, with respect to multiplication and the ordinary addition of vectors, a numerical system which contains the…
Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs…
Over the last decades quaternions have become a crucial and very successful tool for attitude representation in robotics and aerospace. However, there is a major problem that is continuously causing trouble in practice when it comes to…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same…
Quaternions provide a unified algebraic and geometric framework for representing three-dimensional rotations without the singularities that afflict Euler-angle parametrisations. This article develops a pedagogical and conceptual analysis of…
More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis rotations. It is shown here that they…