Higher Dimensions
As has been suggested in earlier sections much of geometric algebra generalises well into higher dimensions. The geometric algebra for the real space ℝn is Cℓn(ℝ), and the subspace of bivectors is Λ2ℝn.
The number of simple bivectors needed to form a general bivector rises with the dimension, so for n odd it is (n - 1) / 2, for n even it is n / 2. So for four and five dimensions only two simple bivectors are needed but three are required for six and seven dimensions. For example in six dimensions with standard basis (e1, e2, e3, e4, e5, e6) the bivector
is the sum of three simple bivectors but no less. As in four dimensions it is always possible to find orthogonal simple bivectors for this sum.
Read more about this topic: Bivector
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