6-orthoplex Hexacross | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 6-polytope |
Family | orthoplex |
Schläfli symbols | {3,3,3,3,4} {3,3,3,3^{1,1}} |
Coxeter-Dynkin diagrams | = |
5-faces | 64 {3^{4}} |
4-faces | 192 {3^{3}} |
Cells | 240 {3,3} |
Faces | 160 {3} |
Edges | 60 |
Vertices | 12 |
Vertex figure | 5-orthoplex |
Petrie polygon | dodecagon |
Coxeter groups | B_{6}, [4,3^{4}] D_{6}, [3^{3,1,1}] |
Dual | 6-cube |
Properties | convex, Hanner polytope |
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.
It has two constructed forms, the first being regular with Schläfli symbol {3^{4},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3^{1,1}} or Coxeter symbol 3_{11}.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract.
This configuration matrix represents the 6-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}
There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C_{6} or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D_{6} or [3^{3,1,1}] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.
Name | Coxeter | Schläfli | Symmetry | Order |
---|---|---|---|---|
Regular 6-orthoplex | {3,3,3,3,4} | [4,3,3,3,3] | 46080 | |
Quasiregular 6-orthoplex | {3,3,3,3^{1,1}} | [3,3,3,3^{1,1}] | 23040 | |
6-fusil | {3,3,3,4}+{} | [4,3,3,3,3] | 7680 | |
{3,3,4}+{4} | [4,3,3,2,4] | 3072 | ||
2{3,4} | [4,3,2,4,3] | 2304 | ||
{3,3,4}+2{} | [4,3,3,2,2] | 1536 | ||
{3,4}+{4}+{} | [4,3,2,4,2] | 768 | ||
3{4} | [4,2,4,2,4] | 512 | ||
{3,4}+3{} | [4,3,2,2,2] | 384 | ||
2{4}+2{} | [4,2,4,2,2] | 256 | ||
{4}+4{} | [4,2,2,2,2] | 128 | ||
6{} | [2,2,2,2,2] | 64 |
Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are
Every vertex pair is connected by an edge, except opposites.
Coxeter plane | B_{6} | B_{5} | B_{4} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B_{3} | B_{2} | |
Graph | |||
Dihedral symmetry | [6] | [4] | |
Coxeter plane | A_{5} | A_{3} | |
Graph | |||
Dihedral symmetry | [6] | [4] |
The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.^{[3]}
2D | 3D | ||
---|---|---|---|
Icosahedron {3,5} = H_{3} Coxeter plane |
6-orthoplex {3,3,3,3^{1,1}} = D_{6} Coxeter plane |
Icosahedron |
6-orthoplex |
This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions as the vertices of a regular icosahedron. This represents a geometric folding of the D_{6} to H_{3} Coxeter groups: : to . On the left, seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. Every pair of vertices of the 6-orthoplex are connected, except opposite ones: 30 edges are shared with the icosahedron, while 30 more edges from the 6-orthoplex project to the interior of the icosahedron. |
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_{k1} series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |
Coxeter diagram |
||||||
Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [[3^{1,3,1}]] = [4,3,3,3,3] |
[3^{2,3,1}] | [3^{3,3,1}] | [3^{4,3,1}] |
Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | 3_{1,-1} | 3_{10} | 3_{11} | 3_{21} | 3_{31} | 3_{41} |
This polytope is one of 63 uniform 6-polytopes generated from the B_{6} Coxeter plane, including the regular 6-cube or 6-orthoplex.