View source for 6-cube

{{Short description|6-dimensional hypercube}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|6-cube<BR>Hexeract
|-
|bgcolor=#ffffff align=center colspan=2|[[File:6-cube graph.svg|280px]]<BR>[[Orthogonal projection]]<BR>inside [[Petrie polygon]]<BR>Orange vertices are doubled, and the center yellow has 4 vertices
|-
|bgcolor=#e7dcc3|Type||Regular [[6-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[hypercube]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|5-faces||12 [[5-cube|{4,3,3,3}]] [[File:5-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||60 [[tesseract|{4,3,3}]] [[File:4-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||160 [[cube|{4,3}]] [[File:3-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||240 [[square (geometry)|{4}]] [[File:2-cube.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||192
|-
|bgcolor=#e7dcc3|Vertices||64
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[5-simplex]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[dodecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||B<sub>6</sub>, [3<sup>4</sup>,4]
|-
|bgcolor=#e7dcc3|Dual||[[6-orthoplex]] [[File:6-orthoplex.svg|25px]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Hanner polytope]]
|}
In [[geometry]], a '''6-cube''' is a six-[[dimension]]al [[hypercube]] with 64 [[Vertex (geometry)|vertices]], 192 [[Edge (geometry)|edge]]s,  240 square [[Face (geometry)|faces]], 160 cubic [[Cell (mathematics)|cells]], 60 [[tesseract]] [[4-face]]s, and 12 [[5-cube]] [[5-face]]s.

It has [[Schläfli symbol]] {4,3<sup>4</sup>}, being composed of 3 [[5-cube]]s around each 4-face. It can be called a '''hexeract''',  a [[portmanteau]] of [[tesseract]] (the ''4-cube'') with ''hex'' for six (dimensions) in [[Greek language|Greek]]. It can also be called a regular '''dodeca-6-tope''' or '''dodecapeton''', being a [[6-polytope|6-dimensional polytope]] constructed from 12 regular [[Facet (geometry)|facet]]s.

== Related polytopes==

It is a part of an infinite family of polytopes, called [[hypercube]]s. The [[Dual polytope|dual]] of a 6-cube can be called a [[6-orthoplex]], and is a part of the infinite family of [[cross-polytope]]s. It is composed of various [[5-cubes]], at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u).<ref>{{Cite web|url=https://www.researchgate.net/publication/361688598|title=A New Six-Dimensional Hyper-Chaotic System}}</ref><ref>{{Cite journal|url=https://www.sciencedirect.com/science/article/pii/S0747717188800105|title=An improved projection operation for cylindrical algebraic decomposition of three-dimensional space - ScienceDirect|journal=Journal of Symbolic Computation |date=February 1988 |volume=5 |issue=1 |pages=141–161 |doi=10.1016/S0747-7171(88)80010-5 |last1=McCallum |first1=Scott }}</ref>

Applying an ''[[Alternation (geometry)|alternation]]'' operation, deleting alternating vertices of the 6-cube, creates another [[uniform polytope]], called a [[6-demicube]], (part of an infinite family called [[demihypercube]]s), which has 12 [[5-demicube]] and 32 [[5-simplex]] facets.

== As a configuration==
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 6-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>

<math>\begin{bmatrix}\begin{matrix}64 & 6 & 15 & 20 & 15 & 6 \\ 2 & 192 & 5 & 10 & 10 & 5 \\ 4 & 4 & 240 & 4 & 6 & 4 \\ 8 & 12 & 6 & 160 & 3 & 3 \\ 16 & 32 & 24 & 8 & 60 & 2 \\ 32 & 80 & 80 & 40 & 10 & 12 \end{matrix}\end{bmatrix}</math>

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a 6-cube centered at the origin and edge length 2 are
: (±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x<sub>0</sub>, x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, x<sub>4</sub>, x<sub>5</sub>) with −1 < x<sub>i</sub> < 1.

== Construction ==

There are three [[Coxeter group]]s associated with the 6-cube, one [[regular polytope|regular]], with the C<sub>6</sub> or [4,3,3,3,3] [[Coxeter group]], and a half symmetry (D<sub>6</sub>) or [3<sup>3,1,1</sup>] Coxeter group. The lowest symmetry construction is based on [[hyperrectangle]]s or [[proprism]]s, [[cartesian product]]s of lower dimensional hypercubes.

{| class="wikitable sortable"
!Name
![[Coxeter diagram|Coxeter]]
![[Schläfli symbol|Schläfli]]
![[Coxeter notation|Symmetry]]
!Order
|- align=center
![[Regular polytope|Regular]] 6-cube
|{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}<BR>{{CDD|node_f1|3|node|3|node|3|node|3|node|4|node}}
|{4,3,3,3,3}
|[4,3,3,3,3]||46080
|- align=center
![[Quasiregular polytope|Quasiregular]] 6-cube
|{{CDD|node_f1|3|node|3|node|3|node|split1|nodes}}
|
|[3,3,3,3<sup>1,1</sup>]||23040
|- align=center
!rowspan=10|[[hyperrectangle]]
|{{CDD|node_1|4|node|3|node|3|node|3|node|2|node_1}}
||{4,3,3,3}×{}||[4,3,3,3,2]||7680
|- align=center
|{{CDD|node_1|4|node|3|node|3|node|2|node_1|4|node}}
||{4,3,3}×{4}||[4,3,3,2,4]||3072
|- align=center
|{{CDD|node_1|4|node|3|node|2|node_1|4|node|3|node}}
||{4,3}<sup>2</sup>||[4,3,2,4,3]||2304
|- align=center
|{{CDD|node_1|4|node|3|node|3|node|2|node_1|2|node_1}}
||{4,3,3}×{}<sup>2</sup>||[4,3,3,2,2]||1536
|- align=center
|{{CDD|node_1|4|node|3|node|2|node_1|4|node|2|node_1}}
||{4,3}×{4}×{}||[4,3,2,4,2]||768
|- align=center
|{{CDD|node_1|4|node|2|node_1|4|node|2|node_1|4|node}}
||{4}<sup>3</sup>||[4,2,4,2,4]||512
|- align=center
|{{CDD|node_1|4|node|3|node|2|node_1|2|node_1|2|node_1}}
||{4,3}×{}<sup>3</sup>||[4,3,2,2,2]||384
|- align=center
|{{CDD|node_1|4|node|2|node_1|4|node|2|node_1|2|node_1}}
||{4}<sup>2</sup>×{}<sup>2</sup>||[4,2,4,2,2]||256
|- align=center
|{{CDD|node_1|4|node|2|node_1|2|node_1|2|node_1|2|node_1}}
||{4}×{}<sup>4</sup>||[4,2,2,2,2]||128
|- align=center
|{{CDD|node_1|2|node_1|2|node_1|2|node_1|2|node_1|2|node_1}}
|{}<sup>6</sup>
|[2,2,2,2,2]||64
|}

== Projections ==
{| class=wikitable
|+ [[orthographic projection]]s
|- align=center
![[Coxeter plane]]
!B<sub>6</sub>
!B<sub>5</sub>
!B<sub>4</sub>
|- align=center
!Graph
|[[File:6-cube t0.svg|150px]]
|[[File:6-cube t0 B5.svg|150px]]
|[[File:4-cube t0.svg|150px]]
|- align=center
![[Dihedral symmetry]]
|[12]
|[10]
|[8]
|- align=center
!Coxeter plane
!Other
!B<sub>3</sub>
!B<sub>2</sub>
|- align=center
!Graph
|[[Image:6-cube column graph.svg|150px]]
|[[File:6-cube t0 B3.svg|150px]]
|[[File:6-cube t0 B2.svg|150px]]
|- align=center
!Dihedral symmetry
|[2]
|[6]
|[4]
|- align=center
!Coxeter plane
!
!A<sub>5</sub>
!A<sub>3</sub>
|- align=center
!Graph
|
|[[File:6-cube t0 A5.svg|150px]]
|[[File:6-cube t0 A3.svg|150px]]
|- align=center
!Dihedral symmetry
|
|[6]
|[4]
|}

{| class="wikitable" width=560
|colspan=2 valign=top align=center|3D Projections
|-
|[[File:Hexeract.ogv|280px]]<br>6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D.
|[[File:6Cube-QuasiCrystal.png|280px]]<br>6-cube [[quasicrystal]] structure orthographically projected<br> to 3D using the [[golden ratio]].
|-
|[[File:Hexeract-q1q4-q2q5-q3q6.gif|280px]]<br>A 3D [[Perspective (graphical)|perspective projection]] of a hexeract undergoing a triple [[rotation]] about the X-W1, Y-W2 and Z-W3 [[Orthogonal coordinates|orthogonal]] [[Rotation plane|planes]].
|}

== Related polytopes==
The 64 vertices of a 6-cube also represent a regular skew 4-polytope {4,3,4 | 4}. Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the [[cubic honeycomb]], {4,3,4}, in 3-dimensions. It has 192 edges, and 192 square faces. Opposite faces fold together into a 4-cycle. Each fold direction adds 1 dimension, raising it into 6-space.

The ''6-cube'' is 6th in a series of [[hypercube]]:
{{Hypercube polytopes}}

This polytope is one of 63 [[uniform 6-polytope]]s generated from the B<sub>6</sub> [[Coxeter plane]], including the regular 6-cube or [[6-orthoplex]].

{{Hexeract family}}

== References ==
<references />
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
* {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta)|o3o3o3o3o4x - ax}}

== External links ==
* {{MathWorld|title=Hypercube|urlname=Hypercube}}
*{{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary: hypercube] Garrett Jones

{{Polytopes}}

[[Category:6-polytopes]]
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