Description
On the previous pages we showed that from a two dimensional vectors we generate an algebra with 4 elements (e, e1, e2and e1^e2) .
| basis | grade |
|---|---|
| e | scalar |
| e1, e2 | vector |
| e1^e2 | bivector (in this case a scalar value) |
We can define a general 'number' in this algebra as a linear sum of these basis:
a + b e1 + c e2+ d e1^e2
We will call this 'number' a multivector, and for 2D vectors is is defined by the 4 scalar values here denoted by a,b,c and d.
In order to understand the properties of this algebra we need to look at the arithmetic rules here.
Such an algebra is the smallest, non-trivial clifford algebra so this may be a good chance to experiment with what we can change.
For instance, we can try changing the following aspects of the algebra:
- Do the basis vectors square to +ve, -ve or zero?
- Do we choose e1^e2 or e2^e1 as the basis bivector?
- Do the components, not on the leading diagonal, anti commute?
These options are discussed on this page.
Cayley Table and Graph
If the basis vectors square to +ve then the Cayley table and graph are as follows:
Cayley Table |
Cayley Graph |
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or as a hypercube:
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There are 2 generators since e1 and e2 are the generators.
Even subalgebra
If we take the even subalgebra (scalar and bivector) then the Cayley table and graph are as follows:
Cayley Table |
Cayley Graph |
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This is isomorphic to the algebra of complex numbers, it has one generator: e12
Comparison of 2D multivector with Quaternion
Clifford algebras based on 2D vectors have 4 'dimensions' so how do they compare with quaternions which also have 4 dimensions? this is discussed on this page.
