4-polytope

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells.
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. Convex 4-polytopes can be "cut and unfolded" as nets in 3-space.
Definition.
A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in which each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i.e. it is not a compound.
The most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube.
Visualisation.
4-polytopes cannot be seen in three-dimensional space due to their extra dimension. Several techniques are used to help visualise them.
Orthogonal projections can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes.
Just as a 3D shape can be projected onto a flat sheet, so a 4-D shape can be projected onto 3-space or even onto a flat sheet. One common projection is a Schlegel diagram which uses stereographic projection of points on the surface of a 3-sphere into three dimensions, connected by straight edges, faces, and cells drawn in 3-space.
Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut "hypersurface" in three dimensions. A sequence of such sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce a smooth animation of these cross sections.
A net of a 4-polytope is composed of polyhedral cells that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane.
Topological characteristics.
The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, and this led to the use of torsion coefficients.
Classification.
Criteria.
Like all polytopes, 4-polytopes may be classified based on properties like "convexity" and "symmetry".
Classes.
The following lists the various categories of 4-polytopes classified according to the criteria above:
Uniform 4-polytope (vertex-transitive):
Other convex 4-polytopes:
Infinite uniform 4-polytopes of Euclidean 3-space (uniform tessellations of convex uniform cells)
Infinite uniform 4-polytopes of hyperbolic 3-space (uniform tessellations of convex uniform cells)
Dual uniform 4-polytope (cell-transitive):
Others:
Abstract regular 4-polytopes:
These categories include only the 4-polytopes that exhibit a high degree of symmetry. Many other 4-polytopes are possible, but they have not been studied as extensively as the ones included in these categories.