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. Geometry of 4D rotations Four-dimensional rotations are of two types: simple rotations and double rotations. Simple rotations A simple rotation about a rotation centre leaves an entire plane through (axis-plane) fixed. Every plane that is completely orthogonal to intersects in a certain point . Each such point is the centre of the 2D rotation induced by in . All these 2D rotations have the same rotation angle . Half-lines from in the axis-plane are not displaced; half-lines from orthogonal to are displaced through ; all other half-lines are displaced through an angle less than . Double rotations For each rotation of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes and each of which is invariant and whose direct sum is all of 4-space. Hence operating on either of these planes produces an ordinary rotation of that plane. For almost all (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles in plane and in plane – both assumed to be nonzero – are different. The unequal rotation angles and satisfying , are almost uniquely determined by . Assuming that 4-space is oriented, then the orientations of the 2-planes and can be chosen consistent with this orientation in two ways. If the rotation angles are unequal (), is sometimes termed a "double rotation". In that case of a double rotation, and are the only pair of invariant planes, and half-lines from the origin in , are displaced through and respectively, and half-lines from the origin not in or are displaced through angles strictly between and . Isoclinic rotations If the rotation angles of a double rotation are equal then there are infinitely many invariant planes instead of just two, and all