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6-cube Hexeract
Orthogonal projection inside Petrie polygon Orange vertices are doubled, and the center yellow has 4 vertices
Type	Regular 6-polytope
Family	hypercube
Schläfli symbol	{4,3⁴}
Coxeter diagram
5-faces	12 {4,3,3,3}
4-faces	60 {4,3,3}
Cells	160 {4,3}
Faces	240 {4}
Edges	192
Vertices	64
Vertex figure	5-simplex
Petrie polygon	dodecagon
Coxeter group	B₆, [3⁴,4]
Dual	6-orthoplex
Properties	convex, Hanner polytope

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

It has Schläfli symbol {4,3⁴}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the 4-cube) with hex for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets.

Related polytopes

It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u).^[1]^[2]

Applying an alternation operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets.

As a configuration

This 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.^[3]^[4]

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

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 (x₀, x₁, x₂, x₃, x₄, x₅) with −1 < x_i < 1.

Construction

There are three Coxeter groups associated with the 6-cube, one regular, with the C₆ or [4,3,3,3,3] Coxeter group, and a half symmetry (D₆) or [3^3,1,1] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.

Name	Schläfli	Symmetry	Order
Regular 6-cube	{4,3,3,3,3}	[4,3,3,3,3]	46080
Quasiregular 6-cube		[3,3,3,3^1,1]	23040
hyperrectangle	{4,3,3,3}×{}	[4,3,3,3,2]	7680
	{4,3,3}×{4}	[4,3,3,2,4]	3072
	{4,3}²	[4,3,2,4,3]	2304
	{4,3,3}×{}²	[4,3,3,2,2]	1536
	{4,3}×{4}×{}	[4,3,2,4,2]	768
	{4}³	[4,2,4,2,4]	512
	{4,3}×{}³	[4,3,2,2,2]	384
	{4}²×{}²	[4,2,4,2,2]	256
	{4}×{}⁴	[4,2,2,2,2]	128
	{}⁶	[2,2,2,2,2]	64

Projections

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	Other	B₃	B₂
Graph
Dihedral symmetry	[2]	[6]	[4]
Coxeter plane		A₅	A₃
Graph
Dihedral symmetry		[6]	[4]

3D Projections
6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D.	6-cube quasicrystal structure orthographically projected to 3D using the golden ratio.
A 3D perspective projection of a hexeract undergoing a triple rotation about the X-W1, Y-W2 and Z-W3 orthogonal 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:

Petrie polygon orthographic projections

Line segment	Square	Cube	4-cube	5-cube	6-cube	7-cube	8-cube	9-cube	10-cube

This polytope is one of 63 uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.

References

^ "(PDF) A New Six-Dimensional Hyper-Chaotic System".
^ "An improved projection operation for cylindrical algebraic decomposition of three-dimensional space - ScienceDirect".
^ Coxeter, Regular Polytopes, sec 1.8 Configurations
^ Coxeter, Complex Regular Polytopes, p.117

Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
Klitzing, Richard. "6D uniform polytopes (polypeta) o3o3o3o3o4x - ax".

External links

Weisstein, Eric W. "Hypercube". MathWorld.
Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Multi-dimensional Glossary: hypercube Garrett Jones

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] "(PDF) A New Six-Dimensional Hyper-Chaotic System".

[2] "An improved projection operation for cylindrical algebraic decomposition of three-dimensional space - ScienceDirect".

[3] Coxeter, Regular Polytopes, sec 1.8 Configurations

[4] Coxeter, Complex Regular Polytopes, p.117

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
+{{Short description|6-dimensional hypercube}}
 {| class="wikitable" align="right" style="margin-left:10px" width="250"
 !bgcolor=#e7dcc3 colspan=2|6-cube<BR>Hexeract
@@ Line 32: / Line 33: @@
 |bgcolor=#e7dcc3|Dual||[[6-orthoplex]] [[File:6-orthoplex.svg|25px]]
 |-
-|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
+|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.
@@ Line 40: / Line 41: @@
 == 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 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_A_New_Six-Dimensional_Hyper-Chaotic_System|title=(PDF) A New Six-Dimensional Hyper-Chaotic System}}</ref><ref>{{Cite web|url=https://www.sciencedirect.com/science/article/pii/S0747717188800105|title=An improved projection operation for cylindrical algebraic decomposition of three-dimensional space - ScienceDirect}}</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==
-The elements of the regular polytopes can be expressed in a [[Configuration_(polytope)|configuration matrix]]. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts ([[f-vector]]s). 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.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>
+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>
@@ Line 108: / Line 109: @@
 |}
-== Images==
+== Projections ==
 {| class=wikitable
 |+ [[orthographic projection]]s
@@ Line 163: / Line 164: @@
 |[[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]].
+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}}

B6 polytopes
β₆	t₁β₆	t₂β₆	t₂γ₆	t₁γ₆	γ₆	t_0,1β₆	t_0,2β₆
t_1,2β₆	t_0,3β₆	t_1,3β₆	t_2,3γ₆	t_0,4β₆	t_1,4γ₆	t_1,3γ₆	t_1,2γ₆
t_0,5γ₆	t_0,4γ₆	t_0,3γ₆	t_0,2γ₆	t_0,1γ₆	t_0,1,2β₆	t_0,1,3β₆	t_0,2,3β₆
t_1,2,3β₆	t_0,1,4β₆	t_0,2,4β₆	t_1,2,4β₆	t_0,3,4β₆	t_1,2,4γ₆	t_1,2,3γ₆	t_0,1,5β₆
t_0,2,5β₆	t_0,3,4γ₆	t_0,2,5γ₆	t_0,2,4γ₆	t_0,2,3γ₆	t_0,1,5γ₆	t_0,1,4γ₆	t_0,1,3γ₆
t_0,1,2γ₆	t_0,1,2,3β₆	t_0,1,2,4β₆	t_0,1,3,4β₆	t_0,2,3,4β₆	t_1,2,3,4γ₆	t_0,1,2,5β₆	t_0,1,3,5β₆
t_0,2,3,5γ₆	t_0,2,3,4γ₆	t_0,1,4,5γ₆	t_0,1,3,5γ₆	t_0,1,3,4γ₆	t_0,1,2,5γ₆	t_0,1,2,4γ₆	t_0,1,2,3γ₆
t_0,1,2,3,4β₆	t_0,1,2,3,5β₆	t_0,1,2,4,5β₆	t_0,1,2,4,5γ₆	t_0,1,2,3,5γ₆	t_0,1,2,3,4γ₆	t_0,1,2,3,4,5γ₆