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.

Rectified 6-simplexes

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.

Rectified 6-simplex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
6. It is also called 0_4,1 for its branching Coxeter-Dynkin diagram, shown as .

Alternate names

Coordinates

The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.

Images

Birectified 6-simplex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
6. It is also called 0_3,2 for its branching Coxeter-Dynkin diagram, shown as .

9-orthoplex

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.

It has two constructed forms, the first being regular with Schläfli symbol {3⁷,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3⁶,3^1,1} or Coxeter symbol 6₁₁.

It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract.

Alternate names

Construction

There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C₉ or [4,3⁷] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D₉ or [3^6,1,1] symmetry group.