So how does this algebra compare with other algebras? Since it has 4 'dimensions' lets compare it with quaternions. Both these algebras implement addition by adding corresponding terms, as usual with this type of algebra, the differences (if any) are in the multiplication rules:
The 2D multivector multiplication table is:
a*b |
b.e | b.e1 | b.e2 | b.e12 |
| a.e | 1 | e1 | e2 | e12 |
| a.e1 | e1 | 1 | e12 | e2 |
| a.e2 | e2 | -e12 | 1 | -e1 |
| a.e12 | e12 | -e2 | e1 | -1 |
The multiplication table for quaternions is:
a*b |
b.1 | b.i | b.j | b.k |
| a.1 | 1 | i | j | k |
| a.i | i | -1 | k | -j |
| a.j | j | -k | -1 | i |
| a.k | k | j | -i | -1 |
These tables are very similar, there is a difference in the 'square' terms, i.e. the terms on the leading diagonal. In the case of 2D multivector there are 3 positives and 1 negative, in the case of the quaternion there is 1 positive and 3 negatives. Otherwise the tables seem very similar, in both cases the real terms commute and the other terms (not on the leading diagonal) anticommute.
Does anyone know if:
- Is there a way to relate these two types?
- Quaternions always have an inverse, do 2D multivectors always have an inverse?
Comparison of 2D multivector with Complex Numbers
We can relate 2D multivectors with complex numbers.
If we let c be a complex number:
c = a + i b
where a and b are the real and imaginary parts.
Assume that we can also represent a complex number by a linear sum of two basis vectors:
x = a e1 + b e2
When we chek this out, using the multipication rules above, this does not quite work. for instance squaring e2 gives +1 not -1.
However if we multiply both sides by e1 we get:
e1 x = a + b e1e2
This has the correct properties, for instance, e1e2 is equivilant to:
e1e2 = √-1
