 6-simplex |  6-orthoplex, 311 |  6-cube (Hexeract) |  221 |
 Expanded 6-simplex |  Rectified 6-orthoplex |  6-demicube 131 (Demihexeract) |  122 |
In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.
A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A vertex is a point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron. A 4-face is a polychoron, and a 5-face is a 5-polytope. Furthermore, the following requirements must be met:
The topology of any given 6-polytope is defined by its Betti numbers and torsion coefficients. [1]
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 6-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. [1]
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients. [1]
6-polytopes may be classified by properties like "convexity" and "symmetry".
Regular 6-polytopes can be generated from Coxeter groups represented by the Schläfli symbol {p,q,r,s,t} with t {p,q,r,s} 5-polytope facets around each cell.
There are only three such convex regular 6-polytopes:
There are no nonconvex regular polytopes of 5 or more dimensions.
For the three convex regular 6-polytopes, their elements are:
| Name | Schläfli symbol | Coxeter diagram | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | Symmetry (order) |
|---|---|---|---|---|---|---|---|---|---|
| 6-simplex | {3,3,3,3,3} |    | 7 | 21 | 35 | 35 | 21 | 7 | A6 (720) |
| 6-orthoplex | {3,3,3,3,4} |  | 12 | 60 | 160 | 240 | 192 | 64 | B6 (46080) |
| 6-cube | {4,3,3,3,3} | | 64 | 192 | 240 | 160 | 60 | 12 | B6 (46080) |
Here are six simple uniform convex 6-polytopes, including the 6-orthoplex repeated with its alternate construction.
| Name | Schläfli symbol(s) | Coxeter diagram(s) | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | Symmetry (order) |
|---|---|---|---|---|---|---|---|---|---|
| Expanded 6-simplex | t0,5{3,3,3,3,3} | | 42 | 210 | 490 | 630 | 434 | 126 | 2×A6 (1440) |
| 6-orthoplex, 311 (alternate construction) | {3,3,3,31,1} |   | 12 | 60 | 160 | 240 | 192 | 64 | D6 (23040) |
| 6-demicube | {3,33,1} h{4,3,3,3,3} |   | 32 | 240 | 640 | 640 | 252 | 44 | D6 (23040) ½B6 |
| Rectified 6-orthoplex | t1{3,3,3,3,4} t1{3,3,3,31,1} | | 60 | 480 | 1120 | 1200 | 576 | 76 | B6 (46080) 2×D6 |
| 221 polytope | {3,3,32,1} |     | 27 | 216 | 720 | 1080 | 648 | 99 | E6 (51840) |
| 122 polytope | {3,32,2} |  or   | 72 | 720 | 2160 | 2160 | 702 | 54 | 2×E6 (103680) |
The expanded 6-simplex is the vertex figure of the uniform 6-simplex honeycomb, . The 6-demicube honeycomb, , vertex figure is a rectified 6-orthoplex and facets are the 6-orthoplex and 6-demicube. The uniform 222 honeycomb,, has 122 polytope is the vertex figure and 221 facets.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In geometry, a 4-polytope 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.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n.
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.
In geometry, a five-dimensional polytope is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.
In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.
In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.