Despite the small dimension, we actually have considerable freedom in the sequence of axis pairs we use; and we also have some freedom in the choice of angles. Thus we find many different conventions employed when three-dimensional rotations are parameterized for physics, or medicine, or chemistry, or other disciplines. When we include the option of world axes or body axes, 24 different sequences are possible. And while some disciplines call any sequence Euler angles, others give different names (Cardano, Tait–Bryan, roll-pitch-yaw) to different sequences. One reason for the large number of options is that, as noted previously, rotations in three dimensions (and higher) do not commute. If we reverse a given sequence of rotations, we get a different outcome. This also implies that we cannot compose two rotations by adding their corresponding angles. Thus Euler angles are not vectors, despite a similarity in appearance as a triplet of numbers. Nested dimensions A rotation matrix such as suggests a rotation matrix, is embedded in the upper left corner: This is no illusion; not just one, but many, copies of -dimensional rotations are found within -dimensional rotations, as subgroups. Each embedding leaves one direction fixed, which in the case of matrices is the rotation axis. For example, we have fixing the -axis, the -axis, and the -axis, respectively. The rotation axis need not be a coordinate axis; if is a unit vector in the desired direction, then where , , is a rotation by angle leaving axis fixed. A direction in -dimensional space will be a unit magnitude vector, which we may consider a point on a generalized sphere, . Thus it is natural to describe the rotation group as combining and . A suitable formalism is the fiber bundle, where for every direction in the base space, , the fiber over it in the total space, , is a copy of the fiber space, , namely the rotations that keep that direction fixed. Thus we can build an rotation matrix by starting with a matrix, aiming its fixed axis on (the ordinary sphere in three-dimensional space), aiming the resulting rotation on , and so on up through . A point on can be selected using numbers, so we again have numbers to describe any rotation matrix. In fact, we can view the sequential angle decomposition, discussed previously, as reversing this process. The composition of Givens rotations brings the first column (and