Biquaternion

The biquaternions are a hyper-complex number system described by William Kingdon Clifford in the second half of the 19th century. Before Clifford, Arthur Cayley had already called the quaternions with complex coefficients (i.e. the set ) biquaternions. ${\displaystyle \mathbb {C} \otimes \mathbb {H} }$

Hamilton Biquaternion

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:

${\displaystyle \mathbf {q} \in \mathbb {C} \otimes \mathbb {H} =\left\{\left.{\begin{pmatrix}w\\x\\y\\z\end{pmatrix}}{\hat {=}}{\begin{pmatrix}w+ix&y+iz\\-y+iz&w-ix\end{pmatrix}}{\hat {=}}{\begin{pmatrix}w&-x&z&-y\\x&w&-y&-z\\-z&y&w&-x\\y&z&x&w\end{pmatrix}}\right\vert w,x,y,z\in \mathbb {C} \right\}}$

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: ${\displaystyle \omega }$ ${\displaystyle \omega }$ ${\displaystyle \omega }$ ${\displaystyle \omega }$

${\displaystyle \mathbf {q} ={\begin{pmatrix}\Re \mathbf {q} _{w}+\omega \,\Im \mathbf {q} _{w}\\\Re \mathbf {q} _{x}+\omega \,\Im \mathbf {q} _{x}\\\Re \mathbf {q} _{y}+\omega \,\Im \mathbf {q} _{y}\\\Re \mathbf {q} _{z}+\omega \,\Im \mathbf {q} _{z}\end{pmatrix}}}$

${\displaystyle \mathbf {q} =\Re \mathbf {q} _{w}+i\,\Re \mathbf {q} _{x}+j\,\Re \mathbf {q} _{y}+k\,\Re \mathbf {q} _{z}+\omega \,\Im \mathbf {q} _{w}+\omega i\,\Im \mathbf {q} _{x}+\omega j\,\Im \mathbf {q} _{y}+\omega k\,\Im \mathbf {q} _{z}}$

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 . ${\displaystyle \omega ^{2}=1}$ ${\displaystyle \omega }$ ${\displaystyle \mathbf {i} _{1}=i}$ ${\displaystyle \mathbf {i} _{2}=j}$ ${\displaystyle \mathbf {i} _{3}=k}$ ${\displaystyle \mathbf {i} _{1}\mathbf {i} _{2}\mathbf {i} _{3}=\omega }$

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: ${\displaystyle \mathbb {C} \otimes \mathbb {H} }$ ${\displaystyle \mathbb {H} \otimes \mathbb {C} }$ ${\displaystyle \mathbb {H} \oplus \mathbb {H} }$

${\displaystyle \mathbf {b} \in \mathbb {H} \otimes \mathbb {C} {\hat {=}}\mathbb {H} \oplus \mathbb {H} \equiv \left\{\left.{\begin{pmatrix}\mathbf {q} \\\mathbf {p} \end{pmatrix}}{\hat {=}}\mathbf {q} +\omega \,\mathbf {p} \right\vert \mathbf {q} ,\mathbf {p} \in \mathbb {H} \right\}}$

Here is the set of complex numbers, the set of quaternions; q and p are quaternions and . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {H} }$ ${\displaystyle \omega ^{2}=1}$

Reshaping

The Clifford biquaternions correspond to the Clifford algebra and form a ring with zero divisors . ${\displaystyle Cl(3,0,\mathbb {R} )}$

The Hamilton biquaternions and the Clifford biquaternions are forms of representation of the biquaternions. A Hamilton biquaternion corresponds to a Clifford biquaternion:

${\displaystyle \mathbf {q} ={\begin{pmatrix}\Re \mathbf {q} _{w}+\omega \,\Im \mathbf {q} _{w}\\\Re \mathbf {q} _{x}+\omega \,\Im \mathbf {q} _{x}\\\Re \mathbf {q} _{y}+\omega \,\Im \mathbf {q} _{y}\\\Re \mathbf {q} _{z}+\omega \,\Im \mathbf {q} _{z}\end{pmatrix}}{\hat {=}}{\begin{pmatrix}\Re \mathbf {q} _{w}\\\Re \mathbf {q} _{x}\\\Re \mathbf {q} _{y}\\\Re \mathbf {q} _{z}\end{pmatrix}}+{\begin{pmatrix}\omega \,\Im \mathbf {q} _{w}\\\omega \,\Im \mathbf {q} _{x}\\\omega \,\Im \mathbf {q} _{y}\\\omega \,\Im \mathbf {q} _{z}\end{pmatrix}}{\hat {=}}{\begin{pmatrix}\Re \mathbf {q} _{w}\\\Re \mathbf {q} _{x}\\\Re \mathbf {q} _{y}\\\Re \mathbf {q} _{z}\end{pmatrix}}+\omega \,{\begin{pmatrix}\Im \mathbf {q} _{w}\\\Im \mathbf {q} _{x}\\\Im \mathbf {q} _{y}\\\Im \mathbf {q} _{z}\end{pmatrix}}}$

Since the octonions are non-associative, they do not have a matrix representation. As a result, biquaternions cannot be converted into octonions.

application

Biquaternions are u. a. used to describe 8-dimensional spaces. Here the time and the spatial dimensions are represented as complex numbers in order to represent the time dilation and the curvature of space .

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. ${\displaystyle \mathbb {R} ^{4}}$ ${\displaystyle \Re q_{w}}$ ${\displaystyle \Im q_{w}}$

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

See also

• Number ranges
  • Integer
  • Rational number
  • Real number
  • Complex number
• Hypercomplex numbers :
  • Octave (octonion)
  • Quaternion
  • Sedenion