5-simplex

Template:Short description Template:Uniform polyteron stat table 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(Template:Sfrac), or approximately 78.46°.

The 5-simplex is a solution to the problem: Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.

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.

As a configuration

This configuration matrix represents the 5-simplex. 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-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.

$\left[\begin{array}{c}\begin{array}{ccccc}6& 5& 10& 10& 5\\ 2& 15& 4& 6& 4\\ 3& 3& 20& 3& 3\\ 4& 6& 4& 15& 2\\ 5& 10& 10& 5& 6\end{array}\end{array}\right]$

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:

$\begin{array}{cc}& (\frac{1}{\sqrt{15}},\frac{1}{\sqrt{10}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{3}},\pm 1)\\ & (\frac{1}{\sqrt{15}},\frac{1}{\sqrt{10}},\frac{1}{\sqrt{6}},-\frac{2}{\sqrt{3}},0)\\ & (\frac{1}{\sqrt{15}},\frac{1}{\sqrt{10}},-\frac{\sqrt{3}}{\sqrt{2}},0,0)\\ & (\frac{1}{\sqrt{15}},-\frac{2\sqrt{2}}{\sqrt{5}},0,0,0)\\ & (-\frac{\sqrt{5}}{\sqrt{3}},0,0,0,0)\end{array}$

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 constructions can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.

Projected images

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

Lower symmetry forms

A lower symmetry form is a 5-cell pyramid {3,3,3}∨( ), with [3,3,3] symmetry order 120, 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. These are seen as vertex figures of truncated regular 6-polytopes, like a truncated 6-cube.

Another form is {3,3}∨{ }, with [3,3,2,1] symmetry order 48, the joining of an orthogonal digon and a tetrahedron, orthogonally offset, with all pairs of vertices connected between. Another form is {3}∨{3}, with [3,2,3,1] symmetry order 36, and extended symmetry [[3,2,3],1], order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between.

The form { }∨{ }∨{ } has symmetry [2,2,1,1], order 8, extended by permuting 3 segments as [3[2,2],1] or [4,3,1,1], order 48.

These are seen in the vertex figures of bitruncated and tritruncated regular 6-polytopes, like a bitruncated 6-cube and a tritruncated 6-simplex. The edge labels here represent the types of face along that direction, and thus represent different edge lengths.

The vertex figure of the omnitruncated 5-simplex honeycomb, Template:CDD, is a 5-simplex with a petrie polygon cycle of 5 long edges. Its symmetry is isomorphic to dihedral group Dih₆ or simple rotation group [6,2]⁺, order 12.

Vertex figures for uniform 6-polytopes

Join	{3,3,3}∨( )	{3,3}∨{ }	{3}∨{3}	{ }∨{ }∨{ }
Symmetry	[3,3,3,1] Order 120	[3,3,2,1] Order 48	[[3,2,3],1] Order 72	[3[2,2],1,1]=[4,3,1,1] Order 48	~[6] or ~[6,2]+ Order 12
Diagram
Polytope	truncated 6-simplex Template:CDD	bitruncated 6-simplex Template:CDD	tritruncated 6-simplex Template:CDD	3-3-3 prism Template:CDD	Omnitruncated 5-simplex honeycomb Template:CDD

Compound

The compound of two 5-simplexes in dual configurations can be seen in this A6 Coxeter plane projection, with a red and blue 5-simplex vertices and edges. This compound has [[3,3,3,3]] symmetry, order 1440. The intersection of these two 5-simplexes is a uniform birectified 5-simplex. Template:CDD = Template:CDD ∩ Template:CDD.

Related uniform 5-polytopes

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.

1_3k dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\tilde{E}}_7$=E7+	${\overline{T}}_8$=E7++
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Symmetry	[3−1,3,1]	[30,3,1]	[31,3,1]	[32,3,1]	[[33,3,1]]	[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 3_k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.

3_k1 dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\tilde{E}}_7$=E7+	${\overline{T}}_8$=E7++
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Symmetry	[3−1,3,1]	[30,3,1]	[[31,3,1]] = [4,3,3,3,3]	[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 2_2k.

2_2k figures of n dimensions

Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	A2A2	A5	E6	${\tilde{E}}_6$=E6+	E6++
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Graph				∞	∞
Name	22,-1	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)

See also

• 11-cell

References

• Template:Cite book
• Coxeter, H.S.M.:
  • Template:Cite book
  • Template:Cite book
    • (Paper 22) Template:Cite journal
    • (Paper 23) Template:Cite journal
    • (Paper 24) Template:Cite journal
• Template:Cite book
• Template:Cite document
  • Template:Cite thesis

External links

• Template:GlossaryForHyperspace
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary