Hypercube (EntityClass, 17)

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>== Number of hypercells in a hypercube ==
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<[#ontology [kind class] [cats Regular Polytope Rotatope Prism]]>
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A '''hypercube''' is an n-dimensional [[polytope]] which is the dual of that dimension's [[cross polytope]]. They exist in all dimensions. They can be represented by the [[bracketopic string]] [a<sub>1</sub>a<sub>2</sub>...a<sub>n</sub>] or by the [[combined Coxeter-Dynkin string]] x4o(3o)*.
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Under the [[elemental naming scheme]], hypercubes are denoted by the ''geo-'' prefix, meaning the classical element of "earth".
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|valign=&quot;top&quot; width=&quot;60%&quot; style=&quot;background-color:#ccccff; text-align:center;&quot; colspan=&quot;6&quot;|'''Dimension of hypercube'''
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== Number of hypercells in a hypercube ==
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|valign=&quot;middle&quot; width=&quot;20%&quot; style=&quot;background-color:#ccccff; text-align:center;&quot; rowspan=&quot;2&quot;|'''Formula'''
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{|style="border: 1px solid; border-color:#808080; border-collapse: collapse;" cellpadding="2" width="100%"
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|valign="top" width="20%" style="background-color:#ccccff; text-align:center;" colspan="2" rowspan="2"|
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|valign="top" width="60%" style="background-color:#ccccff; text-align:center;" colspan="6"|'''Dimension of hypercube'''
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|valign="middle" width="20%" style="background-color:#ccccff; text-align:center;" rowspan="2"|'''Formula'''
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|valign=&quot;middle&quot; width=&quot;10%&quot; style=&quot;background-color:#ccccff; text-align:center;&quot; rowspan=&quot;7&quot;|http://teamikaria.com/dl/53XdjTekuHgkmy3it1dwNHjoxEx6aAvGVcFtlKBiqRtTM2DK.png
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|valign="middle" width="10%" style="background-color:#ccccff; text-align:center;" rowspan="7"|<[#embed [hash 10BDR1GFA3PDYCEP3YJRCCXV4D]]>
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#ddddee; text-align:center;&quot;|2&lt;sup&gt;n&lt;/sup&gt;
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|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|2<sup>n</sup>
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#ddddee; text-align:center;&quot;|n2&lt;sup&gt;n-1&lt;/sup&gt;
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|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|n2<sup>n-1</sup>
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#ddddee; text-align:center;&quot;|n(n-1)2&lt;sup&gt;n-2&lt;/sup&gt;2&lt;sup&gt;-1&lt;/sup&gt;
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|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|n(n-1)2<sup>n-2</sup>2<sup>-1</sup>
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#ddddee; text-align:center;&quot;|n(n-1)(n-2)2&lt;sup&gt;n-3&lt;/sup&gt;6&lt;sup&gt;-1&lt;/sup&gt;
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|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|n(n-1)(n-2)2<sup>n-3</sup>6<sup>-1</sup>
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#eeeeff; text-align:center;&quot;|1
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#eeeeff; text-align:center;&quot;|10
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|valign="top" width="10%" style="background-color:#eeeeff; text-align:center;"|10
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#ddddee; text-align:center;&quot;|n(n-1)(n-2)(n-3)2&lt;sup&gt;n-4&lt;/sup&gt;24&lt;sup&gt;-1&lt;/sup&gt;
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|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|n(n-1)(n-2)(n-3)2<sup>n-4</sup>24<sup>-1</sup>
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#ddddee; text-align:center;&quot;|n/a
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|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|n(n-1)(n-2)(n-3)(n-4)2<sup>n-5</sup>120<sup>-1</sup>
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#ddddee; text-align:center;&quot;|Sum
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|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|Sum
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#eeeeff; text-align:center;&quot;|9
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|valign=&quot;top&quot; width=&quot;10%&quot; style=&quot;background-color:#ddddee; text-align:center;&quot;|3&lt;sup&gt;n&lt;/sup&gt;
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|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|3<sup>n</sup>
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Number of k-cubes in an n-cube: 2&lt;sup&gt;n-k&lt;/sup&gt;n!/(k!(n-k)!)
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Number of k-cubes in an n-cube: 2<sup>n-k</sup>n!/(k!(n-k)!)
{{Hypercubes| }}
{{Hypercubes| }}
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Latest revision as of 14:08, 15 March 2014

A hypercube is an n-dimensional polytope which is the dual of that dimension's cross polytope. They exist in all dimensions. They can be represented by the bracketopic string [a1a2...an] or by the combined Coxeter-Dynkin string x4o(3o)*.

Under the elemental naming scheme, hypercubes are denoted by the geo- prefix, meaning the classical element of "earth".

Number of hypercells in a hypercube

Dimension of hypercube Formula
0 1 2 3 4 5
(image) 0 1 2 4 8 16 32 2n
1 0 1 4 12 32 80 n2n-1
2 0 0 1 6 24 80 n(n-1)2n-22-1
3 0 0 0 1 8 40 n(n-1)(n-2)2n-36-1
4 0 0 0 0 1 10 n(n-1)(n-2)(n-3)2n-424-1
5 0 0 0 0 0 1 n(n-1)(n-2)(n-3)(n-4)2n-5120-1
Sum 1 3 9 27 81 243 3n

Number of k-cubes in an n-cube: 2n-kn!/(k!(n-k)!)


Hypercubes
point • digon • square • cube • geochoron • geoteron • geopeton
Retrieved from "http://hi.gher.space/wiki/Hypercube"

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