Search results
Results from the WOW.Com Content Network
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world.
Stereographic projection of the duocylinder's ridge (see below), as a flat torus.The ridge is rotating about the xw-plane.. The duocylinder, also called the double cylinder or the bidisc, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii r 1 and r 2:
The regular complex polytope 4 {4} 2, , in has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 4 {4} 2 has 16 vertices, and 8 4-edges. Its symmetry is 4 [4] 2, order 32. It also has a lower symmetry construction, , or 4 {}× 4 {}, with symmetry 4 [2] 4, order 16. This is the symmetry if the red and blue 4-edges are ...
The spherinder can be seen as the volume between two parallel and equal solid 2-spheres (3-balls) in 4-dimensional space, here stereographically projected into 3D.. In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball (or solid 2-sphere) of radius r 1 and a line segment of length 2r 2:
The space P n (K) is often called the projective space of dimension n over K, or the projective n-space, since all projective spaces of dimension n are isomorphic to it (because every K vector space of dimension n + 1 is isomorphic to K n+1). The elements of a projective space P(V) are commonly called points.
Quaternionic projective space of dimension n is usually denoted by and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line is homeomorphic to the 4-sphere.
The term polychoron (plural polychora, adjective polychoric), from the Greek roots poly ("many") and choros ("room" or "space") and was advocated [10] by Norman Johnson and George Olshevsky in the context of uniform polychora (4-polytopes), and their related 4-dimensional symmetry groups.
A point in 4-dimensional space with Cartesian coordinates (u, x, y, z) may be represented by a quaternion P = u + xi + yj + zk. A left-isoclinic rotation is represented by left-multiplication by a unit quaternion Q L = a + bi + cj + dk. In matrix-vector language this is