Hypercube (EntityClass, 17)

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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...a_n] 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	Formula
0	1	2	4	8	16	32	2ⁿ
1	0	1	4	12	32	80	n2^n-1
2	0	0	1	6	24	80	n(n-1)2^n-22^-1
3	0	0	0	1	8	40	n(n-1)(n-2)2^n-36^-1
4	0	0	0	0	1	10	n(n-1)(n-2)(n-3)2^n-424^-1
5	0	0	0	0	0	1	n(n-1)(n-2)(n-3)(n-4)2^n-5120^-1
Sum	1	3	9	27	81	243	3ⁿ

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

Hypercubes

point • digon • square • cube • geochoron • geoteron • geopeton

Pages in this category (7)

Square

@@ Line 1: / Line 1: @@
-A '''hypercube''' is an [[n-dimensional]] analog to a [[cube]]. Hypercubes include the [[point (object)|point]], [[line (object)|line]], [[square]], [[cube]] and [[tesseract]].
+<[#ontology [kind class] [cats Regular Polytope Rotatope Prism]]>
+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)*.
-== Equations ==
+Under the [[elemental naming scheme]], hypercubes are denoted by the ''geo-'' prefix, meaning the classical element of "earth".
-=== Number of hypercells in a hypercube ===
+== Number of hypercells in a hypercube ==
-{|cellpadding="14" cellspacing="0" width="100%"
-|valign="top" width="50%"|
 {|style="border: 1px solid; border-color:#808080; border-collapse: collapse;" cellpadding="2" width="100%"
 |valign="top" width="20%" style="background-color:#ccccff; text-align:center;" colspan="2" rowspan="2"|
@@ Line 18: / Line 18: @@
 |valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|5
 |-
-|valign="middle" width="10%" style="background-color:#ccccff; text-align:center;" rowspan="7"|http://invhost.com/share/dimhyc_rotated.png
+|valign="middle" width="10%" style="background-color:#ccccff; text-align:center;" rowspan="7"|<[#embed [hash 10BDR1GFA3PDYCEP3YJRCCXV4D]]>
 |valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|0
 |valign="top" width="10%" style="background-color:#eeeeff; text-align:center;"|1
@@ Line 71: / Line 71: @@
 |valign="top" width="10%" style="background-color:#eeeeff; text-align:center;"|0
 |valign="top" width="10%" style="background-color:#eeeeff; text-align:center;"|1
-|valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|n/a
+|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>
 |-
 |valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|Sum
@@ Line 82: / Line 82: @@
 |valign="top" width="10%" style="background-color:#ddddee; text-align:center;"|3<sup>n</sup>
 |}
-Number of k-cubes in an n-cube: 2<sup>n-k</sup>n!/(k!(n-k!))
+Number of k-cubes in an n-cube: 2<sup>n-k</sup>n!/(k!(n-k)!)
-|}
-[[Category:Shapes]]
+{{Hypercubes| }}