Regular 5-orthoplex (pentacross) | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 5-polytope |
Family | orthoplex |
Schläfli symbol | {3,3,3,4} {3,3,3^{1,1}} |
Coxeter-Dynkin diagrams | |
4-faces | 32 {3^{3}} |
Cells | 80 {3,3} |
Faces | 80 {3} |
Edges | 40 |
Vertices | 10 |
Vertex figure | 16-cell |
Petrie polygon | decagon |
Coxeter groups | BC_{5}, [3,3,3,4] D_{5}, [3^{2,1,1}] |
Dual | 5-cube |
Properties | convex |
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
It has two constructed forms, the first being regular with Schläfli symbol {3^{3},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3^{1,1}} or Coxeter symbol 2_{11}.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.
This configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}
Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are
There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C_{5} or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D_{5} or [3^{2,1,1}] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.
Name | Coxeter diagram | Schläfli symbol | Symmetry | Order | Vertex figure(s) |
---|---|---|---|---|---|
regular 5-orthoplex | {3,3,3,4} | [3,3,3,4] | 3840 | ||
Quasiregular 5-orthoplex | {3,3,3^{1,1}} | [3,3,3^{1,1}] | 1920 | ||
5-fusil | |||||
{3,3,3,4} | [4,3,3,3] | 3840 | |||
{3,3,4}+{} | [4,3,3,2] | 768 | |||
{3,4}+{4} | [4,3,2,4] | 384 | |||
{3,4}+2{} | [4,3,2,2] | 192 | |||
2{4}+{} | [4,2,4,2] | 128 | |||
{4}+3{} | [4,2,2,2] | 64 | |||
5{} | [2,2,2,2] | 32 |
Coxeter plane | B_{5} | B_{4} / D_{5} | B_{3} / D_{4} / A_{2} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B_{2} | A_{3} | |
Graph | |||
Dihedral symmetry | [4] | [4] |
The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection. |
2_{k1} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Finite | Euclidean | Hyperbolic | ||||||||
n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
Coxeter group |
E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||
Coxeter diagram |
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Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [[3^{1,2,1}]] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||
Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||
Graph | - | - | |||||||||
Name | 2_{−1,1} | 2_{01} | 2_{11} | 2_{21} | 2_{31} | 2_{41} | 2_{51} | 2_{61} |
This polytope is one of 31 uniform 5-polytopes generated from the B_{5} Coxeter plane, including the regular 5-cube and 5-orthoplex.