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Concept
Rotations in 4-dimensional Euclidean space
Summary
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.
In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise.
A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation.
Four-dimensional rotations are of two types: simple rotations and double rotations.
A simple rotation R about a rotation centre O leaves an entire plane A through O (axis-plane) fixed. Every plane B that is completely orthogonal to A intersects A in a certain point P. For each such point P is the centre of the 2D rotation induced by R in B. All these 2D rotations have the same rotation angle α.
Half-lines from O in the axis-plane A are not displaced; half-lines from O orthogonal to A are displaced through α; all other half-lines are displaced through an angle less than α.
For each rotation R of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes A and B each of which is invariant and whose direct sum A ⊕ B is all of 4-space. Hence R operating on either of these planes produces an ordinary rotation of that plane. For almost all R (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles α in plane A and β in plane B – both assumed to be nonzero – are different. The unequal rotation angles α and β satisfying −π < α, β < π are almost uniquely determined by R. Assuming that 4-space is oriented, then the orientations of the 2-planes A and B can be chosen consistent with this orientation in two ways. If the rotation angles are unequal (α ≠ β), R is sometimes termed a "double rotation".
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