# 6-cube > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/6-cube > Markdown URL: https://mediated.wiki/source/6-cube.md > Source: https://en.wikipedia.org/wiki/6-cube > Source revision: 1356812269 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|6-dimensional hypercube}} {| class="wikitable" style="float:right; margin-left:10px; width:290px" !style="background:#e7dcc3" colspan=2|6-cube
Hexeract |- |style="background:#fff; text-align:center" colspan=2|280px
[Orthogonal projection](/source/Orthogonal_projection)
inside [Petrie polygon](/source/Petrie_polygon)
Orange vertices are doubled, and the center yellow has 4 vertices |- |style="background:#e7dcc3"|Type||Regular [6-polytope](/source/6-polytope) |- |style="background:#e7dcc3"|Family||[hypercube](/source/hypercube) |- |style="background:#e7dcc3"|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)|| {4,34} |- |style="background:#e7dcc3"|[Coxeter diagram](/source/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}](/source/5-cube) 25px|class=skin-invert |- |style="background:#e7dcc3"|4-faces||60 [{4,3,3}](/source/tesseract) 25px|class=skin-invert |- |style="background:#e7dcc3"|Cells||160 [{4,3}](/source/cube) 25px|class=skin-invert |- |style="background:#e7dcc3"|Faces||240 [{4}](/source/square_(geometry)) 25px|class=skin-invert |- |style="background:#e7dcc3"|Edges||192 |- |style="background:#e7dcc3"|Vertices||64 |- |style="background:#e7dcc3"|[Vertex figure](/source/Vertex_figure)||[5-simplex](/source/5-simplex) |- |style="background:#e7dcc3"|[Petrie polygon](/source/Petrie_polygon)||[dodecagon](/source/dodecagon) |- |style="background:#e7dcc3"|[Coxeter group](/source/Coxeter_group)||B6, [34,4] |- |style="background:#e7dcc3"|Dual||[6-orthoplex](/source/6-orthoplex) 25px|class=skin-invert |- |style="background:#e7dcc3"|Properties||[convex](/source/Convex_polytope), [Hanner polytope](/source/Hanner_polytope) |} In [geometry](/source/geometry), a '''6-cube''' is a six-[dimension](/source/dimension)al [hypercube](/source/hypercube) with 64 [vertices](/source/Vertex_(geometry)), 192 [edge](/source/Edge_(geometry))s, 240 square [faces](/source/Face_(geometry)), 160 cubic [cells](/source/Cell_(mathematics)), 60 [tesseract](/source/tesseract) [4-face](/source/4-face)s, and 12 [5-cube](/source/5-cube) [5-face](/source/5-face)s. It has [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {4,34}, being composed of 3 [5-cube](/source/5-cube)s around each 4-face. It can be called a '''hexeract''', a [portmanteau](/source/portmanteau) of [tesseract](/source/tesseract) (the ''4-cube'') with ''hex'' for six (dimensions) in [Greek](/source/Greek_language). It can also be called a regular '''dodeca-6-tope''' or '''dodecapeton''', being a [6-dimensional polytope](/source/6-polytope) constructed from 12 regular [facet](/source/Facet_(geometry))s. == Related polytopes == It is a part of an infinite family of polytopes, called [hypercube](/source/hypercube)s. The [dual](/source/Dual_polytope) of a 6-cube can be called a [6-orthoplex](/source/6-orthoplex), and is a part of the infinite family of [cross-polytope](/source/cross-polytope)s. It is composed of various [5-cubes](/source/5-cubes), at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u).{{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 }}{{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 }} Applying an ''[alternation](/source/Alternation_(geometry))'' operation, deleting alternating vertices of the 6-cube, creates another [uniform polytope](/source/uniform_polytope), called a [6-demicube](/source/6-demicube), (part of an infinite family called [demihypercube](/source/demihypercube)s), which has 12 [5-demicube](/source/5-demicube) and 32 [5-simplex](/source/5-simplex) facets. == As a configuration == This [configuration matrix](/source/Regular_4-polytope) 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}} \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](/source/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 < xi < 1. == Construction == There are three [Coxeter group](/source/Coxeter_group)s associated with the 6-cube, one [regular](/source/regular_polytope), with the C6 or [4,3,3,3,3] [Coxeter group](/source/Coxeter_group), and a half symmetry (D6) or [33,1,1] Coxeter group. The lowest symmetry construction is based on [hyperrectangle](/source/hyperrectangle)s or [proprism](/source/proprism)s, [cartesian product](/source/cartesian_product)s of lower dimensional hypercubes. {| class="wikitable sortable" !Name ![Coxeter](/source/Coxeter_diagram) ![Schläfli](/source/Schl%C3%A4fli_symbol) ![Symmetry](/source/Coxeter_notation) !Order |- align=center ![Regular](/source/Regular_polytope) 6-cube |{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}
{{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](/source/Quasiregular_polytope) 6-cube |{{CDD|node_f1|3|node|3|node|3|node|split1|nodes}} | |[3,3,3,31,1]||23040 |- align=center !rowspan=10|[hyperrectangle](/source/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 projection](/source/Orthographic_projection)s |- align=center ![Coxeter plane](/source/Coxeter_plane) !B6 !B5 !B4 |- align=center !Graph |150px|class=skin-invert |150px|class=skin-invert |150px|class=skin-invert |- align=center ![Dihedral symmetry](/source/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
6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D. |280px
6-cube [quasicrystal](/source/quasicrystal) structure orthographically projected
to 3D using the [golden ratio](/source/golden_ratio). |- |280px
A 3D [perspective projection](/source/Perspective_(graphical)) of a hexeract undergoing a triple [rotation](/source/rotation) about the X-W1, Y-W2 and Z-W3 [orthogonal](/source/Orthogonal_coordinates) [planes](/source/Rotation_plane). |} == 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](/source/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](/source/hypercube)s: {{Hypercube polytopes}} This polytope is one of 63 [uniform 6-polytope](/source/uniform_6-polytope)s generated from the B6 [Coxeter plane](/source/Coxeter_plane), including also the regular [6-orthoplex](/source/6-orthoplex). {{Hexeract family}} == 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}} Category:6-polytopes Category:Articles containing video clips --- Adapted from the Wikipedia article [6-cube](https://en.wikipedia.org/wiki/6-cube) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/6-cube?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.