Higher Dimensions
It is possible to define parameters analogous to the Euler angles in dimensions higher than three.
The number of degrees of freedom of a rotation matrix is always less than the dimension of the matrix squared. That is, the elements of a rotation matrix are not all completely independent. For example, the rotation matrix in dimension 2 has only one degree of freedom, since all four of its elements depend on a single angle of rotation. A rotation matrix in dimension 3 (which has nine elements) has three degrees of freedom, corresponding to each independent rotation, for example by its three Euler angles or a magnitude one (unit) quaternion.
In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The 4x4 rotation matrices have therefore 6 out of 16 independent components.
Any set of 6 parameters that define the rotation matrix could be considered an extension of Euler angles to dimension 4.
In general, the number of euler angles in dimension D is quadratic in D; since any one rotation consists of choosing two dimensions to rotate between, the total number of rotations available in dimension D is, which for D=2,3,4 yields .
Read more about this topic: Euler Angles
Famous quotes containing the words higher and/or dimensions:
“Many, no doubt, are well disposed, but sluggish by constitution and habit, and they cannot conceive of a man who is actuated by higher motives than they are. Accordingly they pronounce this man insane, for they know that they could never act as he does, as long as they are themselves.”
—Henry David Thoreau (1817–1862)
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)