Төртөлшемді политоптар және олардың түрлері

Four dimensional geometric object with flat sides
+ Graphs of the six convex regular 4 polytopes{3,3,3}{3,3,4}{4,3,3}5 cellPentatope4 simplex16 cellOrthoplex4 orthoplex8 cellTesseract4 cube{3,4,3}{3,3,5}{5,3,3}24 cellOctaplex600 cellTetraplex120 cellDodecaplex
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 4 polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853. 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.

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 convex regular 4 polytopes are the four dimensional analogues of the Platonic solids. The most familiar 4 polytope is the tesseract or hypercube, the 4D analogue of the cube. 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 within the same radius. The 4 simplex (5 cell) is the limit smallest case, and the 120 cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering.