Quaternions

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Quaternions are an extension to the real numbers, similar to the complex numbers. While the real numbers are extended to the complex numbers by adding a number i such that i² = -1, quaternions are extended by defining i, j and k such that i² = j² = k² = ijk = -1. A quaternion then is a number of the form A + Bi + Cj + Dk, where A, B, C and D are real numbers.

Unlike real or complex numbers, multiplication of quaternions is not commutative: ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. The quaternions are an example of a [skew field]?, an algebraic stucture similar to a field except for commutativity of multiplication. In particular, multiplication is still associative and avery non-zero element has a unique inverse. The quaternions, along with the complex and real numbers, are the only finite dimensional skew fields over the field of real numbers.

Quaternions were discovered by [William Rowan Hamilton]?. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. He could not do so for 3-dimensions, but 4-dimensions produce quaternions. (According to a story he told, he was out walking one day when the solution in the form of equation i² = j² = k² = ijk = -1 suddenly occurred to him; he then proceeded to carve this equation into the side of a bridge!)

The set of quaternions of absolute value 1 forms a 3-dimensional sphere S³ and a group. This group acts by conjugation on the copy of R³ consisting of quaternions with real part equal to zero: it is not hard to see that the conjugation by a unit quaternion of real part cos t is a rotation by an angle 2t, the axis of the rotation being the direction of the imaginary part. Thus, S³ is the double cover of the group SO(3) of real orthogonal 3x3 matrices of determinant 1; it is isomorphic to SU(2), the group of complex unitary 2x2 matrices of determinant 1.

Let A be the set of quaternions of the form a + bi + cj + dk where a, b, c and d are either all integers or all rational numbers with odd numerator and denominator 2. The set A is a ring and a lattice. There are 24 unit quaternions in this ring and they are the vertices of a [regular polytope]? called {3,4,3} in Schlafli's notation.

Quaternions are sometimes used in computer graphics (and associated geometric analysis) to represent the orientation of an object in 3d space. The advantages are: non singular representation (compared with [Euler angles]? for example), more compact (and faster) than matrices.