5-simplex Hexateron (hix)
Type	uniform 5-polytope
Schläfli symbol	{34}
Coxeter diagram
4-faces	6	6{3,3,3}
Cells	15	15{3,3}
Faces	20	20{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,isogonalregular,self-dual

Infive-dimensionalgeometry, a 5-simplex is a self-dualregular5-polytope. It has sixvertices, 15edges, 20 trianglefaces, 15 tetrahedralcells, and 65-cellfacets. It has adihedral angle of cos⁻¹(⁠1/5⁠), 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.

It can also be called ahexateron, orhexa-5-tope, as a 6-facetted polytope in 5-dimensions. Thenamehexateron is derived fromhexa- for having sixfacets andteron (withter- being a corruption oftetra-) for having four-dimensional facets.

By Jonathan Bowers, a hexateron is given the acronymhix.^[1]

Thisconfiguration 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.^[2][3]

Thehexateron can be constructed from a5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.

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

The vertices of the5-simplex can be more simply positioned on ahyperplane 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 the6-orthoplex orrectified 6-cube respectively.

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 ofSchlegel diagram 5D to 4D of hexateron.

A lower symmetry form is a5-cell pyramid {3,3,3}∨( ), with [3,3,3] symmetry order 120, constructed as a5-cell base in a 4-spacehyperplane, and anapex pointabove the hyperplane. The fivesides of the pyramid are made of 5-cell cells. These are seen asvertex figures of truncated regular6-polytopes, like atruncated 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 thevertex figures ofbitruncated and tritruncated regular 6-polytopes, like abitruncated 6-cube and atritruncated 6-simplex. The edge labels here represent the types of face along that direction, and thus represent different edge lengths.

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

Vertex figures foruniform 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	bitruncated 6-simplex	tritruncated 6-simplex	3-3-3 prism	Omnitruncated 5-simplex honeycomb

The compound of two 5-simplexes in dual configurations can be seen in this A6Coxeter 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 uniformbirectified 5-simplex. = ∩.

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

1_3k dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\displaystyle {\tilde{E}}_7}$=E7+	${\displaystyle {\overline{T}}_8}$=E7++
Coxeter diagram
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 byCoxeter as 3_k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedraldihedron.

3_k1 dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\displaystyle {\tilde{E}}_7}$=E7+	${\displaystyle {\overline{T}}_8}$=E7++
Coxeter diagram
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 ofn dimensions

Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	A2A2	A5	E6	${\displaystyle {\tilde{E}}_6}$=E6+	E6++
Coxeter diagram
Graph				∞	∞
Name	22,-1	220	221	222	223

The regular 5-simplex is one of 19uniform polytera based on the [3,3,3,3]Coxeter group, all shown here in A₅Coxeter planeorthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

A5 polytopes
t0	t1	t2	t0,1	t0,2	t1,2	t0,3
t1,3	t0,4	t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4
t0,2,4	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,1,2,3,4

• 11-cell

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• Multi-dimensional Glossary

• 5-polytopes

• Articles with short description
• Short description is different from Wikidata