6-cube

{{Short description|6-dimensional hypercube}} {| class="wikitable" style="float:right; margin-left:10px; width:290px" !style="background:#e7dcc3" colspan=2|6-cube<BR>Hexeract |- |style="background:#fff; text-align:center" colspan=2|280px<BR>Orthogonal projection<BR>inside Petrie polygon<BR>Orange vertices are doubled, and the center yellow has 4 vertices |- |style="background:#e7dcc3"|Type||Regular 6-polytope |- |style="background:#e7dcc3"|Family||hypercube |- |style="background:#e7dcc3"|Schläfli symbol|| {4,3⁴} |- |style="background:#e7dcc3"|Coxeter diagram||{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}} |- |style="background:#e7dcc3"|5-faces||12 {4,3,3,3} 25px|class=skin-invert |- |style="background:#e7dcc3"|4-faces||60 {4,3,3} 25px|class=skin-invert |- |style="background:#e7dcc3"|Cells||160 {4,3} 25px|class=skin-invert |- |style="background:#e7dcc3"|Faces||240 {4} 25px|class=skin-invert |- |style="background:#e7dcc3"|Edges||192 |- |style="background:#e7dcc3"|Vertices||64 |- |style="background:#e7dcc3"|Vertex figure||5-simplex |- |style="background:#e7dcc3"|Petrie polygon||dodecagon |- |style="background:#e7dcc3"|Coxeter group||B₆, [3⁴,4] |- |style="background:#e7dcc3"|Dual||6-orthoplex 25px|class=skin-invert |- |style="background:#e7dcc3"|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).<ref>{{cite book |last1=Mehdi |first1=Sadiq A. |last2=Ali |first2=Zaydon L. |title=2019 International Engineering Conference (IEC) |chapter=A New Six-Dimensional Hyper-Chaotic System |date=2019 |pages=211–215 |doi=10.1109/IEC47844.2019.8950634 |isbn=978-1-7281-4377-4 }}</ref><ref>{{cite journal |last1=McCallum |first1=Scott |title=An improved projection operation for cylindrical algebraic decomposition of three-dimensional space |journal=Journal of Symbolic Computation |date=February 1988 |volume=5 |issue=1–2 |pages=141–161 |doi=10.1016/S0747-7171(88)80010-5 }}</ref>

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.{{sfnp|Coxeter|1973|p=12|at=Sec. 1.8 Configurations}}{{sfnp|Coxeter|1991|p=117}}

<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 (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.

{| class="wikitable sortable" !Name !Coxeter !Schläfli !Symmetry !Order |- align=center !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 6-cube |{{CDD|node_f1|3|node|3|node|3|node|split1|nodes}} | |[3,3,3,3^1,1]||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}²||[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}×{}²||[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}³||[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}×{}³||[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}²×{}²||[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}×{}⁴||[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}} |{}⁶ |[2,2,2,2,2]||64 |}

== Projections == {| class=wikitable |+ Orthographic projections |- align=center !Coxeter plane !B₆ !B₅ !B₄ |- align=center !Graph |150px|class=skin-invert |150px|class=skin-invert |150px|class=skin-invert |- align=center !Dihedral symmetry |[12] |[10] |[8] |- align=center !Coxeter plane !Other !B₃ !B₂ |- align=center !Graph |150px|class=skin-invert |150px|class=skin-invert |150px|class=skin-invert |- align=center !Dihedral symmetry |[2] |[6] |[4] |- align=center !Coxeter plane ! !A₅ !A₃ |- align=center !Graph | |150px|class=skin-invert |150px|class=skin-invert |- align=center !Dihedral symmetry | |[6] |[4] |}

{| class="wikitable" width=560 |colspan=2 valign=top align=center|3D Projections |- |280px<br>6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D. |280px<br>6-cube quasicrystal structure orthographically projected<br> to 3D using the golden ratio. |- |280px<br>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 hypercubes: {{Hypercube polytopes}}

This polytope is one of 63 uniform 6-polytopes generated from the B₆ Coxeter plane, including also the regular 6-orthoplex. {{Hexeract family}}

== References == <references /> * {{cite book |last=Coxeter |first=H.S.M. |author-link=H. S. M. Coxeter |title=Regular Polytopes |title-link=Regular Polytopes (book) |year=1973 |edition=3rd |publisher=Dover Publications |location=New York |page=296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n'' ≥ 5) |isbn=0-486-61480-8}} * {{cite book | last=Coxeter | first=H.S.M. | title=Regular Complex Polytopes | publisher=Cambridge University Press | year=1991 | orig-year=1974 | isbn=0-521-39490-2 }} * {{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}}

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