5-simplex

5-simplex
Hexateron (hix)
Type uniform 5-polytope
Schläfli symbol {34}
Coxeter diagram
4-faces66 {3,3,3}
Cells1515 {3,3}
Faces2020 {3}
Edges 15
Vertices 6
Vertex figure
5-cell
Coxeter group A5, [34], order 720
Dual self-dual
Base point (0,0,0,0,0,1)
Circumradius 0.645497
Properties convex, isogonal regular, self-dual

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.

Alternate names

It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions. The name hexateron is derived from hexa- for having six facets and teron (with ter- being a corruption of tetra-) for having four-dimensional facets.

By Jonathan Bowers, a hexateron is given the acronym hix.[1]

Regular hexateron cartesian coordinates

The hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.

The Cartesian coordinates for the vertices of an origin-centered regular hexateron having edge length 2 are:

\left(\sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)
\left(\sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)
\left(\sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)
\left(\sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)
\left(-\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)

The vertices of the 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,0,1) or (0,1,1,1,1,1). These construction can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.

Projected images

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [3]

Stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of hexateron.

Related uniform 5-polytopes

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.

13k dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 {\tilde{E}}_{7}=E7+ {\bar{T}}_8=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[3<sup>3,3,1</sup>]] [34,3,1]
Order 48 720 23,040 2,903,040
Graph - -
Name 13,-1 130 131 132 133 134

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.

3k1 dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 {\tilde{E}}_{7}=E7+ {\bar{T}}_8=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [[3<sup>1,3,1</sup>]] [32,3,1] [33,3,1] [34,3,1]
Order 48 720 46,080 2,903,040
Graph - -
Name 31,-1 310 311 321 331 341

The 5-simplex, as 220 polytope is first in dimensional series 22k.

22k figures of n dimensions
Space Finite Euclidean Hyperbolic
n 5 6 7 8
Coxeter
group
A5 E6 {\tilde{E}}_{6}=E6+ E6++
Coxeter
diagram
Graph
Name 220 221 222 223

The regular 5-simplex is one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

Other forms

The 5-simplex can also be considered a 5-cell pyramid, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells.

Notes

  1. Klitzing, (x3o3o3o3o - hix)

References

External links

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