Four-dimensional Space - Geometry

Geometry

See also: Rotations in 4-dimensional Euclidean space

The geometry of 4-dimensional space is much more complex than that of 3-dimensional space, due to the extra degree of freedom.

Just as in 3 dimensions there are polyhedra made of two dimensional polygons, in 4 dimensions there are polychora (4-polytopes) made of polyhedra. In 3 dimensions there are 5 regular polyhedra known as the Platonic solids. In 4 dimensions there are 6 convex regular polychora, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform polychora, analogous to the 13 semi-regular Archimedean solids in three dimensions.

Regular polytopes in four dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)

A4	BC4		F4	H4
5-cell	tesseract	16-cell	24-cell	120-cell	600-cell

In 3 dimensions, a circle may be extruded to form a cylinder. In 4 dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps"), and a cylinder may be extruded to obtain a cylindrical prism. The Cartesian product of two circles may be taken to obtain a duocylinder. All three can "roll" in 4-dimensional space, each with its own properties.

In 3 dimensions, curves can form knots but surfaces cannot (unless they are self-intersecting). In 4 dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction, but 2-dimensional surfaces can form non-trivial, non-self-intersecting knots in 4-dimensional space. Because these surfaces are 2-dimensional, they can form much more complex knots than strings in 3-dimensional space can. The Klein bottle is an example of such a knotted surface. Another such surface is the real projective plane.