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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:
[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.
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
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 ...
The regular convex 4-polytopes are the four-dimensional analogues of the Platonic solids in three dimensions and the convex regular polygons in two dimensions. Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size.
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The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space V is generalized to Grassmannian manifold, which is parametrizing higher-dimensional subspaces (of some fixed dimension) of V. sequence of subspaces More generally flag manifold is the space of flags, i.e., chains of linear subspaces of V.
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