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n-cube rotation
In general, rotation is a planar phenomenon requiring two dimensions to operate. Any additional dimensions in the space that the rotating object is embedded in manifests itself as a stationary set.
No rotation is possible in 1 dimension. An object in 1 dimension cannot rotate without leaving that 1-dimensional space.
In 2 dimensions, both dimensions are used for rotation, leaving a 0-dimensional stationary point. Hence, an object in 2 dimensions rotate about a point. The rotational axis commonly associated with 2-dimensional rotation actually lies outside of the 2-dimensional space itself, and thus is merely an artifact of anthropocentric bias toward 3-dimensional space. Hence, rotation in 2 dimensions is more properly understood as rotation about a center of rotation. Rotations in 2 dimensions are uniquely identified by the center of rotation and the rate of rotation.No rotation is possible in 1 dimension. An object in 1 dimension cannot rotate without leaving that 1-dimensional space.
In 3 dimensions, objects rotate about an axis, a stationary line, since there is one dimension "left over" as the other two participate in the rotation. The rotational axis is peculiar to odd-numbered dimensions. Rotations in 3 dimensions are uniquely identified by the axis of rotation and the rate of rotation.
Rotation in 4 dimensions are of two kinds: plane rotations and composite rotations. A plane rotation has a stationary plane which an object may rotate "around". This is because two dimensions participate in the rotation while the other two are stationary. Objects in 4 dimensions can also rotate independently in these two leftover dimensions, resulting in a composite rotation composed of two plane rotations at two independent rates of rotation. These composite rotations have a stationary point, just as in 2 dimensions. Hence, rotation in 4 dimensions are identified by a center of rotation, and two rates of rotation (planar rotation being a special case where one of the rates is zero).
In 5 dimensions, rotations have either a rotational axis or a rotational 3-space. With the former, there are two independent rates of rotation, just as in 4 dimensions. With the latter, there is 1 rate of rotation (2 dimensions participating in the rotation, and 3 dimensions forming the stationary 3-space).
In general, in n dimensions, if n is odd then rotations have an axis, and there are (n-1)/2 possible simultaneous plane rotations around that axis. If n is even, then rotations have stationary points (rotational centers), with n/2 possible simultaneous plane rotations. Each possible plane rotation has its own rate of rotation.
• This page was last changed on 15 February 2010, at 13:32.
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