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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.
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 ...
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. These are fitted together along their respective faces (face-to-face) in a regular fashion, forming the surface of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of ...
The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content [5] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest.
In E. coli, the IIB Chb is a monomer. Two IIB Chb monomers associate with the IIA Chb dimer. The structure of the IIB domain of the Chb porter has been determined both by NMR and by X-ray crystallography. It exhibits an α/β doubly wound superfold. This is different from the structure of the IIB Glc and IIB Man domains.
The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of Michael Freedman () implies that the homeomorphism type of the manifold only depends on this intersection form, and on a / invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kirby ...
A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.
In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by surgery theory. The Whitney trick requires 2+1 dimensions (2 space, 1 time), hence the two Whitney disks of surgery theory require 2+2+1=5 dimensions.