geometric algebra for the real space Rn is Cln(R), and the subspace of bivectors is ⋀2Rn. The number of simple bivectors needed to form a general bivector rises with the dimension, so for n odd it is , for n even it is . 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. Rotations in higher dimensions As in three and four dimensions rotors are generated by the exponential map, so is the rotor generated by bivector B. Simple rotations, that take place in a plane of rotation around a fixed blade of dimension are generated by simple bivectors, while other bivectors generate more complex rotations which can be described in terms of the simple bivectors they are sums of, each related to a plane of rotation. All bivectors can be expressed as the sum of orthogonal and commutative simple bivectors, so rotations can always be decomposed into a set of commutative rotations about the planes associated with these bivectors. The group of the rotors in n dimensions is the spin group, Spin(n). One notable feature, related to the number of simple bivectors and so rotation planes, is that in odd dimensions every rotation has a fixed axis – it is misleading to call it an axis of rotation as in higher dimensions rotations are taking place in multiple planes orthogonal to it. This is related to bivectors, as bivectors in odd dimensions decompose into the same number of bivectors as the even dimension below, so have the same number of planes, but one extra dimension. As each plane generates rotations in two dimensions in odd dimensions there must be one dimension, that is an axis, that is not being rotated. Bivectors are also related to the rotation matrix in n dimensions. As in three dimensions the characteristic equation of the matrix can be solved to find the eigenvalues. In odd dimensions this has one real root, with eigenvector the fixed axis, and in even dimensions it has no real roots, so either all or all but one of the roots are complex conjugate pairs. Each