Numerical Calculations in Clifford Algebra

7Quaternions

Quaternions were constructed by Hamilton (1844) as a generalisation of complex numbers from two to three dimensions. They were used by Maxwell (1873a, 1873b) to write his equations of electromagnetism.

7.1 Theory

Quaternions are represented in Clifford algebra by identifying the unit quaternions with particular Clifford units:

(7.1)

in a context where the signature , as deduced by both Grassmann (1995) and Clifford (1878), is all . Quaternions form a particular subset of three‐dimensional Clifford numbers. Whereas three‐dimensional Clifford numbers have four grades and eight components, quaternions retain only two of the grades (0 and 2) and four of the components.

The unit quaternions obey the rules for multiplication (Hamilton 1844):

(7.2)

These rules are honoured by the Clifford algebra when Grassmann's central multiplication is used.

Table 6.1 lists the operations ...