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.
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.^{[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.