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The rotation is completely specified by specifying the axis planes and the angles of rotation about them. Without loss of generality, these axis planes may be chosen to be the uz - and xy-planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation. In 4D space, the Hopf angles {ξ 1, η, ξ 2} parameterize the 3 ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
Einstein's concept of spacetime has a Minkowski structure based on a non-Euclidean geometry with three spatial dimensions and one temporal dimension, rather than the four symmetric spatial dimensions of Schläfli's Euclidean 4D space. Single locations in Euclidean 4D space can be given as vectors or 4-tuples, i.e., as ordered lists of numbers ...
1951, A. C. Hurley, Finite rotation groups and crystal classes in four dimensions, Proceedings of the Cambridge Philosophical Society, vol. 47, issue 04, p. 650 [1] 1962 A. L. MacKay Bravais Lattices in Four-dimensional Space [2] 1964 Patrick du Val, Homographies, quaternions and rotations, quaternion-based 4D point groups
The two rotation planes span four-dimensional space, so every point in the space can be specified by two points, one on each of the planes. A double rotation has two angles of rotation, one for each plane of rotation. The rotation is specified by giving the two planes and two non-zero angles, α and β (if either angle is zero the rotation is ...
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations , rotations , reflections , and glide reflections (see below § Classification ).
If a rotation of Minkowski space is in a space-like plane, then this rotation is the same as a spatial rotation in Euclidean space. By contrast, a rotation in a plane spanned by a space-like dimension and a time-like dimension is a hyperbolic rotation , and if this plane contains the time axis of the reference frame, is called a "Lorentz boost".
A stereographic projection of a Clifford torus performing a simple rotation Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together. In geometric topology , the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles S 1