Quaternion

A quaternion is an expression of the form a + bi + cj + dk, where a, b, c, d are real numbers and i, j, k are imaginary units satisfying i² = j² = k² = ijk = −1. The set of all quaternions forms a non-commutative division algebra over the reals, denoted by H in honor of Hamilton.¹

Unlike complex numbers, multiplication of quaternions is not commutative: ij = k but ji = −k. This property is crucial for their geometric applications. The conjugate of a quaternion is obtained by negating the imaginary parts, and the norm is the square root of the sum of squares of the components.

Unit quaternions (those with norm 1) provide a compact and numerically stable way to represent rotations in 3D space. A rotation by angle θ around a unit axis (x, y, z) is represented by the quaternion cos(θ/2) + sin(θ/2)(xi + yj + zk).²

Quaternions avoid the gimbal lock problem of Euler angles and require fewer operations than rotation matrices. They are widely used in computer graphics for character animation, in robotics for orientation control, and in spacecraft attitude determination, such as in the Apollo program's guidance computer.³

Hamilton conceived quaternions on October 16, 1843, while walking along the Royal Canal in Dublin; he carved the fundamental relation into a stone of Brougham Bridge.⁴ The discovery was a milestone in algebra, leading to the development of vector analysis by Josiah Willard Gibbs and Oliver Heaviside.

Quaternions are the only finite-dimensional real division algebra besides the reals and complex numbers, as proven by Frobenius in 1877. They also form a normed division algebra, one of only four (the others being reals, complex numbers, and octonions).

Quaternions have surprising connections to number theory: the Lipschitz integers (quaternions with integer components) and Hurwitz integers (with half-integer components) are used to prove Lagrange's four-square theorem, which states every natural number is the sum of four squares.⁵

In physics, quaternions have been used to formulate special relativity, and they appear in the study of the Dirac equation. The quaternion group Q8 is a non-abelian group of order 8 that appears in group theory and crystallography. Quaternions also have applications in computer vision for pose estimation and in the analysis of protein structures.

Unit quaternion

A quaternion with norm 1, used to represent rotations.

Gimbal lock

Loss of one degree of freedom in a three-axis mechanism, avoided by quaternions.

Octonions

An eight-dimensional non-associative algebra extending quaternions.