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| {{about|the mathematical concept|the film|Cube 2: Hypercube|the computer architecture|Connection Machine}}
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| {| align=right class=wikitable
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| |+ [[Perspective projection]]s
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| |[[File:Hexahedron.svg|190px]]
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| |[[File:Hypercube.svg|190px]]
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| ![[Cube]] (3-cube)
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| ![[Tesseract]] (4-cube)
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| |}
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| In [[geometry]], a '''hypercube''' is an ''n''-dimensional analogue of a [[Square (geometry)|square]] (''n'' = 2) and a [[cube]] (''n'' = 3). It is a [[Closed set|closed]], [[Compact space|compact]], [[Convex polytope|convex]] figure whose 1-[[skeleton (topology)|skeleton]] consists of groups of opposite [[parallel (geometry)|parallel]] [[line segment]]s aligned in each of the space's [[dimension]]s, [[perpendicular]] to each other and of the same length. A unit hypercube's longest diagonal in n-dimensions is equal to <math>\sqrt{n}</math>.
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| An ''n''-dimensional hypercube is also called an '''n-cube'''. The term "measure polytope" is also used, notably in the work of [[H.S.M. Coxeter]] (originally from Elte, 1912<ref>{{Cite document | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912 | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->}} Chapter IV, five dimensional semiregular polytope [http://www.amazon.com/Semiregular-Polytopes-Hyperspaces-Emanuel-Lodewijk/dp/141817968X]</ref>), but it has now been superseded.
| | DVR organizes your shows on a menu and it is possible to place them into folders that you could delete or lock with password protection, also. Direc - TV DVR by Tivo, [http://www.jeremymazzonetto.com/additifs-alimentaires/index.php?title=Want_to_Know_More_About_Wireless_Security_Camera_Systems permits] you to record one, or even two, shows in the same time.<br><br>How To Find A Good Massage Therapist ' 5 Answers Any Good Therapist Should Know. Look for chapters on a variety of writing such as travel, humor and business writing. [http://demo.o3dstaging.nl/handlers/printcontent.cfm?GroupID=4&ContentID=15&ThisPageURL=http%3A//cctvdvrreviews.com&EntryCode=7615 wired cctv dvr security camera system] cctv dvr system uk Stating the problem will be the first step to be 16 channel dvr card considered when writing an action plan to correct problems. Run on sentences are really easy to spot because they're two independent sentences sandwiched together. |
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| The hypercube is the special case of a [[hyperrectangle]] (also called an ''orthotope'').
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| A '''unit hypercube''' is a hypercube whose side has length one unit. Often, the hypercube whose corners (or '''vertices''') are the 2<sup>''n''</sup> points in ''R<sup>n</sup>'' with coordinates equal to 0 or 1 is called '''"the" unit hypercube'''.
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| == Construction ==
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| [[Image:Dimension levels.svg|thumb|left|350px|A diagram showing how to create a tesseract from a point.]]
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| :'''0''' – A point is a hypercube of dimension zero.
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| :'''1''' – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.
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| :'''2''' – If one moves this line segment its length in a [[perpendicular]] direction from itself; it sweeps out a 2-dimensional square.
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| :'''3''' – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.
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| :'''4''' – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit [[tesseract]]).
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| This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a [[Minkowski sum]]: the ''d''-dimensional hypercube is the Minkowski sum of ''d'' mutually perpendicular unit-length line segments, and is therefore an example of a [[zonotope]].
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| The 1-[[Skeleton (topology)|skeleton]] of a hypercube is a [[hypercube graph]].
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| == Coordinates ==
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| A unit hypercube of ''n'' dimensions is the [[convex hull]] of the points given by all sign permutations of the [[Cartesian coordinates]] <math>\left(\pm \frac{1}{2}, \pm \frac{1}{2}, \cdots, \pm \frac{1}{2}\right)</math>. It has an edge length of 1 and an ''n''-dimensional volume of 1.
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| An ''n''-dimensional hypercube is also often regarded as the convex hull of all sign permutations of the coordinates <math>(\pm 1, \pm 1, \cdots, \pm 1)</math>. This form is often chosen due to ease of writing out the coordinates. Its edge length is 2, and its ''n''-dimensional volume is 2<sup>n</sup>.
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| == Related families of polytopes ==
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| The hypercubes are one of the few families of [[regular polytope]]s that are represented in any number of dimensions.
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| The '''hypercube (offset)''' family is one of three [[regular polytope]] families, labeled by [[Coxeter]] as ''γ<sub>n</sub>''. The other two are the hypercube dual family, the '''[[cross-polytope]]s''', labeled as ''β<sub>n</sub>'', and the '''[[simplex|simplices]]''', labeled as ''α<sub>n</sub>''. A fourth family, the [[hypercubic honeycomb|infinite tessellations of hypercubes]], he labeled as ''δ<sub>n</sub>''.
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| Another related family of semiregular and [[uniform polytope]]s is the '''[[demihypercube]]s''', which are constructed from hypercubes with alternate vertices deleted and [[simplex]] facets added in the gaps, labeled as ''hγ<sub>n</sub>''.
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| == Elements == | |
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| Every n-cube of n > 0 is composed of elements, or n-cubes of a lower dimension, on the (n-1)-dimensional surface on the parent hypercube.
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| A side is any element of (n-1) dimension of the parent hypercube. A hypercube of dimension n has 2n sides (a 1-dimensional line has 2 end points; a 2-dimensional square has 4 sides or edges; a 3-dimensional cube has 6 2-dimensional faces; a 4-dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is <math>2^{n}</math> (a cube has <math>2^{3}</math> vertices, for instance).
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| A simple formula to calculate the number of ''"n-2"''-faces in an ''n''-dimensional hypercube is: <math>2n^{2}-2n</math>
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| The number of ''m''-dimensional hypercubes (just referred to as m-cube from here on) on the boundary of an ''n''-cube is
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| :<math> E_{m,n} = 2^{n-m}{n \choose m} </math>, where <math>{n \choose m}=\frac{n!}{m!\,(n-m)!}</math> and ''n''! denotes the [[factorial]] of ''n''.
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| For example, the boundary of a 4-cube (n=4) contains 8 cubes (3-cubes), 24 squares (2-cubes), 32 lines (1-cubes) and 16 vertices (0-cubes).
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| This identity can be proved by combinatorial arguments; each of the <math>2^n</math> vertices defines a vertex in
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| a <math>m</math>-dimensional boundary. There are <math>{n \choose m}</math> ways of choosing which lines ("sides") that defines the subspace that the boundary is in. But, each side is counted <math>2^m</math> times since it has that many vertices, we need to divide with this number. Hence the identity above.
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| These numbers can also be generated by the linear [[recurrence relation]]
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| :<math>E_{m,n} = 2E_{m,n-1} + E_{m-1,n-1} \!</math>, with <math>E_{0,0} = 1 \!</math>, and undefined elements = 0.
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| For example, extending a square via its 4 vertices adds one extra line (edge) per vertex, and also adds the final second square, to form a cube, giving <math>E_{1,3} \!</math> = 12 lines in total.
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| {| class="wikitable"
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| |+
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| Hypercube elements <math>E_{m,n} \!</math> {{OEIS|A013609}}
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| |-
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| !
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| !
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| !
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| ! m
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| ! 0
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| ! 1
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| ! 2
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| ! 3
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| ! 4
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| ! 5
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| ! 6
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| ! 7
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| ! 8
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| ! 9
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| ! 10
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| |-
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| ! [[polytope|n]]
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| ! γ<sub>n</sub>
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| ! n-cube
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| ! Names<BR>[[Schläfli symbol]]<BR>[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
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| ! [[Vertex (geometry)|Vertices]]
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| ! [[Edge (geometry)|Edges]]
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| ! [[Face (geometry)|Faces]]
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| ! [[Cell (geometry)|Cells]]<BR>(3-faces)
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| ! ''4''-faces
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| ! ''5''-faces
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| ! ''6''-faces
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| ! ''7''-faces
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| ! ''8''-faces
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| ! ''9''-faces
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| ! ''10''-faces
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| |-
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| ! [[0-polytope|0]]
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| ! γ<sub>0</sub>
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| | 0-cube
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| | Point<BR>-
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| | 1
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| |-
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| ! [[1-polytope|1]]
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| ! γ<sub>1</sub>
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| | 1-cube
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| | [[Line segment]]<BR>{}<BR>{{CDD|node_1}}
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| | 2
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| | 1
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| |-
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| ! [[2-polytope|2]]
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| ! γ<sub>2</sub>
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| | 2-cube
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| | [[Square (geometry)|Square]]<BR>'''Tetragon'''<BR>{4}<BR>{{CDD|node_1|4|node}}
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| | 4
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| | 4
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| | 1
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| |-
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| ! [[3-polytope|3]]
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| ! γ<sub>3</sub>
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| | 3-cube
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| | [[Cube]]<BR>'''Hexahedron'''<BR>{4,3}<BR>{{CDD|node_1|4|node|3|node}}
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| | 8
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| | 12
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| | 6
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| | 1
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| |-
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| ! [[4-polytope|4]]
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| ! γ<sub>4</sub>
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| | 4-cube
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| | [[Tesseract]]<BR>'''Octachoron'''<BR>{4,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node}}
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| | 16
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| | 32
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| | 24
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| | 8
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| | 1
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| |
| |
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| |-
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| ! [[5-polytope|5]]
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| ! γ<sub>5</sub>
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| | 5-cube
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| | [[Penteract]]<BR>'''Decateron'''<BR>{4,3,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node}}
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| | 32
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| | 80
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| | 80
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| | 40
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| | 10
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| | 1
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| |
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| |-
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| ! [[6-polytope|6]]
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| ! γ<sub>6</sub>
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| | 6-cube
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| | [[Hexeract]]<BR>'''Dodecapeton'''<BR>{4,3,3,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}
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| | 64
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| | 192
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| | 240
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| | 160
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| | 60
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| | 12
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| | 1
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| |
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| |-
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| ! [[7-polytope|7]]
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| ! γ<sub>7</sub>
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| | 7-cube
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| | [[Hepteract]]<BR>'''Tetradeca-7-tope'''<BR>{4,3,3,3,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}
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| | 128
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| | 448
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| | 672
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| | 560
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| | 280
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| | 84
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| | 14
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| | 1
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| |
| |
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| |-
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| ! [[8-polytope|8]]
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| ! γ<sub>8</sub>
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| | 8-cube
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| | [[Octeract]]<BR>'''Hexadeca-8-tope'''<BR>{4,3,3,3,3,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| | 256
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| | 1024
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| | 1792
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| | 1792
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| | 1120
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| | 448
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| | 112
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| | 16
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| | 1
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| |
| |
| |
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| |-
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| ! [[9-polytope|9]]
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| ! γ<sub>9</sub>
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| | 9-cube
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| | [[Enneract]]<BR>'''Octadeca-9-tope'''<BR>{4,3,3,3,3,3,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| | 512
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| | 2304
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| | 4608
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| | 5376
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| | 4032
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| | 2016
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| | 672
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| | 144
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| | 18
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| | 1
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| |
| |
| |-
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| ! [[10-polytope|10]]
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| ! γ<sub>10</sub>
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| | 10-cube
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| | [[10-cube|Dekeract]]<BR>'''icosa-10-tope'''<BR>{4,3,3,3,3,3,3,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| |1024
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| |5120
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| |11520
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| |15360
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| |13440
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| |8064
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| |3360
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| |960
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| |180
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| |20
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| |1
| |
| |}
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| === Graphs === | |
| An '''n-cube''' can be projected inside a regular ''2n''-gonal polygon by a [[Petrie_polygon#The_hypercube_and_orthoplex_families|skew orthogonal projection]], shown here from the line segment to the 12-cube.
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| {| class=wikitable
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| |+ [[Petrie polygon]] [[Orthographic projection]]s
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| |- align=center
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| |[[File:1-simplex t0.svg|160px]]<BR>[[Line segment]]
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| |[[File:2-cube.svg|160px]]<BR>[[Square (geometry)|Square]]
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| |[[File:3-cube graph.svg|160px]]<BR>[[Cube]]
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| |[[File:4-cube graph.svg|160px]]<BR>4-cube ([[tesseract]])
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| |- align=center
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| |[[File:5-cube graph.svg|160px]]<BR>[[5-cube]] ([[penteract]])
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| |[[File:6-cube graph.svg|160px]]<BR>[[6-cube]] ([[hexeract]])
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| |[[File:7-cube graph.svg|160px]]<BR>[[7-cube]] ([[hepteract]])
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| |[[File:8-cube.svg|160px]]<BR>[[8-cube]] ([[octeract]])
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| |- align=center
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| |[[File:9-cube.svg|160px]]<BR>[[9-cube]] ([[enneract]])
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| |[[File:10-cube.svg|160px]]<BR>[[10-cube]] ([[dekeract]])
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| |[[File:11-cube.svg|160px]]<BR>[[11-cube]] ([[hendekeract]])
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| |[[File:12-cube.svg|160px]]<BR>[[12-cube]] ([[dodekeract]])
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| |}
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| [[File:8-cell.gif|right|thumb|256px|Projection of a [[rotation|rotating]] tesseract.]]
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| ==Relation to ''n''-simplices==
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| The graph of the ''n''-hypercube's edges is [[isomorphism|isomorphic]] to the [[Hasse diagram]] of the (''n''-1)-[[simplex]]'s [[Convex polytope#The_face_lattice|face lattice]]. This can be seen by orienting the ''n''-hypercube so that two opposite vertices lie vertically, corresponding to the (''n''-1)-simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (''n''-1)-simplex's facets (''n''-2 faces), and each vertex connected to those vertices maps to one of the simplex's ''n''-3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices.
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| This relation may be used to generate the face lattice of an (''n-1'')-simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive.
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| ==See also==
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| * [[Hyperoctahedral group]], the symmetry group of the hypercube
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| * [[Hypersphere]]
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| * [[Simplex]]
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| * [[MIMD#Hypercube_interconnection_network|Hypercube interconnection network]] of computer architecture
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{cite journal
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| |authorlink = Jonathan Bowen
| |
| |last = Bowen
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| |first = J. P.
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| |url =https://web.archive.org/web/20080630081518/www.jpbowen.com/publications/ndcubes.html
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| |title = Hypercubes
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| |journal = [[Practical Computing]]
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| |volume = 5
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| |issue = 4
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| |pages = 97–99
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| |date=April 1982
| |
| }}
| |
| * {{cite book
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| |authorlink = Harold Scott MacDonald Coxeter
| |
| |last = Coxeter
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| |first = H. S. M.
| |
| |title = [[Regular Polytopes (book)|Regular Polytopes]]
| |
| |edition = 3rd
| |
| |publisher = Dover
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| |year = 1973
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| |pages = 123
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| |isbn = 0-486-61480-8
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| }} p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n'' ≥ 5)
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| * {{cite book
| |
| |first = Frederick J.
| |
| |last = Hill
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| |author2 = Gerald R. Peterson
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| |title = Introduction to Switching Theory and Logical Design: Second Edition
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| |publisher = John Wiley & Sons
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| |place = NY
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| |isbn = 0-471-39882-9
| |
| }} Cf Chapter 7.1 "Cubical Representation of Boolean Functions" wherein the notion of "hypercube" is introduced as a means of demonstrating a distance-1 code ([[Gray code]]) as the vertices of a hypercube, and then the hypercube with its vertices so labelled is squashed into two dimensions to form either a [[Veitch diagram]] or [[Karnaugh map]].
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| == External links ==
| |
| * {{MathWorld|title=Hypercube|urlname=Hypercube}}
| |
| * {{MathWorld|title=Hypercube graphs|urlname=HypercubeGraph}}
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| * {{GlossaryForHyperspace | anchor=Measure | title=Measure polytope}}
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| * [http://www.4d-screen.de/related-space/ www.4d-screen.de] (Rotation of 4D – 7D-Cube)
| |
| * ''[http://demonstrations.wolfram.com/RotatingAHypercube/ Rotating a Hypercube]'' by Enrique Zeleny, [[Wolfram Demonstrations Project]].
| |
| * [http://dogfeathers.com/java/hyprcube.html Stereoscopic Animated Hypercube]
| |
| * [http://www.cs.sjsu.edu/~rucker/hypercube.htm Rudy Rucker and Farideh Dormishian's Hypercube Downloads]
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| {{Dimension topics}}
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| {{Polytopes}}
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| [[Category:Articles with inconsistent citation formats]]
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| [[Category:Multi-dimensional geometry]]
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| [[Category:Cubes]]
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