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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 ...
Cyclical Projections. Chapter XI. A Tessaractic Figure and its Projections. Appendices A. 100 Names used for Plane Space. B. 216 Names used for Cubic Space. C. 256 Names used for Tessaractic Space. D. List of Colours, Names and Symbols. E. A Theorem in Four-Space. F. Exercises on Shapes of Three Dimensions. G. Exercises on Shapes of Four ...
2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere into 3-space.. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue), and hypermeridians (gr
A projective space is a topological space, as endowed with the quotient topology of the topology of a finite dimensional real vector space. Let S be the unit sphere in a normed vector space V , and consider the function π : S → P ( V ) {\displaystyle \pi :S\to \mathbf {P} (V)} that maps a point of S to the vector line passing through it.
[4] The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron. Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling.
Four-dimensional space, the concept of a fourth spatial dimension; Spacetime, the unification of time and space as a four-dimensional continuum; Minkowski space, the mathematical setting for special relativity