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...

rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation; see rotations in 4-dimensional Euclidean space...

is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also...

A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because...

three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are...

also a Euclidean geometric system with four real Cartesian coordinates. Cayley used quaternions to study rotations in 4-dimensional Euclidean space. At mid-century...

the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are...

(higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension, including six found in 4-dimensional space. Schläfli's...

a Euclidean space of dimension n, En (Euclidean line, E; Euclidean plane, E2; Euclidean three-dimensional space, E3) form a real coordinate space of...

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention...

In mathematics and theoretical physics, a pseudo-Euclidean space of signature (k, n-k) is a finite-dimensional real n-space together with a non-degenerate...

as ordinary rotations of the four-dimensional Euclidean sphere. The four-dimensional spacetime can be visualized as a four-dimensional space, with each...

unit quaternions represent a rotation in 4D space (see Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations). The set of all unit quaternions...

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from...

describing more complex rotations in four-dimensional space and higher dimensions, where they can be used to break down the rotations into simpler parts....

coordinate system axis. Rotations in 4-dimensional Euclidean space can be seen as the composition of two 2-dimensional rotations in completely orthogonal...

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space...

system to another. In geometry the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation...

composition of two 2-dimensional rotations in completely orthogonal planes. The 16-cell is a simple frame in which to observe 4-dimensional rotations, because each...

Elfrinkhof described rotations in 4-dimensional Euclidean space. A system of national secretaries was announced in the AMS Bulletin in 1899: Alexander McAulay...

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent...

that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical...

(cis(x) = cos(x) + i sin(x)) Quaternions and spatial rotation Rotations in 4-dimensional Euclidean space Turn (geometry) Mukunda, N.; Simon, R.; Sudarshan...

"pure" rotations as linear maps in a vector space equipped with Euclidean structure, not as maps of points of a corresponding affine space. In other words...

associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric...

left-right symmetry. Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries...

In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating...

points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms, and all three-dimensional Euclidean spaces are considered identical...

x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} in k-dimensional space ℝk, the elements of their Euclidean distance matrix A are given by squares of distances...

Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions...