The biquaternions described by Arthur Cayley are quaternions whose elements are complex numbers. This can be represented - by converting the quaternion - whereby the quaternion is represented as a 4-vector, as a complex 2 × 2 matrix or as a 4 × 4 matrix:
The Biquaternionen are thus an 8-dimensional hypercomplex number system with the units 1, i j, k , i, j, k. A biquaternion q can thus e.g. B. be represented as follows:
Here i, j and k are the units of the quaternions . It is also true that i, j and k commute . The components w, x, y and z represent the respective dimensions that are represented by the quaternion. The compound is formed by , , and .
Clifford Biquaternion
The Clifford biquaternions arise from the idea of replacing the complex numbers in the Hamilton biquaternions with a divided complex number. This can be achieved by transforming the expression into . This can be thought of as creating a “complex number with quaternions” instead of a “quaternion with complex numbers”. Alternatively, the biquaternions can be formed as the direct sum of the quaternions with themselves, that is. For the biquaternion b this can be defined as follows:
Here is the set of complex numbers, the set of quaternions; q and p are quaternions and .
The Hamilton biquaternions and the Clifford biquaternions are forms of representation of the biquaternions. A Hamilton biquaternion corresponds to a Clifford biquaternion:
Since the octonions are non-associative, they do not have a matrix representation. As a result, biquaternions cannot be converted into octonions.
A simpler application is the use of the biquaternion to represent a straight line ( vertex ) in 4-dimensional space ( ), with the real part representing the support vector and the imaginary part representing the direction vector . For use in animated 3D computer graphics, the time t is used for the factor - the factor is not required and is therefore set to zero.
credentials
WK Clifford; Preliminary Sketch of Biquaternions. ; Proc. London Math. Soc. 4, 381-395, 1873
WR Hamilton; Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method ; Hodges and Smith, Dublin, 1853
E. Study, Of Movement and Reallocation ; Math. Ann. 39, 441-566, 1891.
van der Waerden, BL A History of Algebra from al-Khwarizmi to Emmy Noether ; Springer-Verlag, pages 188-189, New York, 1985. ISBN 038713610X