Clifford biquaternion

The Biquaternionen are a hypercomplex number system, which was described by William Kingdon Clifford in the second half of the 19th century. Before Clifford Arthur Cayley had already (ie the amount ) denotes the quaternions with complex coefficients as Biquaternionen.

Hamilton biquaternion

The Biquaternionen described by Arthur Cayley are quaternions whose elements are complex numbers. This can - be displayed, wherein the quaternion is represented as a 4- vector, a complex 2 x 2 matrix or 4 × 4 matrix - by converting the quaternion:

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 be represented by, for example, as follows:

Clifford biquaternion

The Clifford - Biquaternionen caused by the idea of ​​replacing the complex numbers in the Hamilton Biquaternionen by a split complex number. This is achieved by forming in the expression. This can be thought of as that is called a " complex number with quaternions " forms instead of a " quaternion with complex numbers ". Alternatively, one can imagine the Biquaternionen as a direct sum of quaternions with itself, ie, form. For biquaternion b This can be defined as follows:

This is the set of complex numbers, the amount of quaternions; q and p are quaternions and.

Transformation

The Clifford - Biquaternionen correspond to the Clifford algebra and form a ring with zero divisors.

The Hamilton Biquaternionen and Clifford Biquaternionen are representations of Biquaternionen. A Hamilton biquaternion corresponds to a Clifford - biquaternion:

In addition Biquaternionen can be converted using the following calculation rules in octonions:

This is true:

Application

Biquaternionen are inter alia used to describe eight - dimensional spaces can. In this case, the time and spatial dimensions are represented as complex numbers to represent the space and time dilation curvature.

A simpler application, the use of the Biquaternions to represent a straight line ( vertex ) in the four -dimensional space (), where the real part of the support vector and the imaginary part represents the direction vector. For use in the animated 3D computer graphics, the time t is in this case used for the factor - the factor is not needed and therefore set to zero.

Credentials

  • W. K. Clifford; Preliminary Sketch of Biquaternions. ; Proc. London Math Soc. 4, 381-395, 1873
  • W. R. Hamilton; Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method; Hodges and Smith, Dublin, 1853
  • E. Study of the movement and rerouting of; 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
Clifford-Algebra Integer Sedenion
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