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== Generalized hypercubes == Regular [[Complex polytope#Regular complex polytopes|complex polytopes]] can be defined in [[Complex number|complex]] [[Hilbert space]] called ''generalized hypercubes'', γ{{supsub|''p''|''n''}} = <sub>''p''</sub>{4}<sub>2</sub>{3}...<sub>2</sub>{3}<sub>2</sub>, or {{CDD|pnode_1|4|node|3}}..{{CDD|3|node|3|node}}. Real solutions exist with ''p'' = 2, i.e. γ{{supsub|2|''n''}} = γ<sub>''n''</sub> = <sub>2</sub>{4}<sub>2</sub>{3}...<sub>2</sub>{3}<sub>2</sub> = {4,3,..,3}. For ''p'' > 2, they exist in <math>\mathbb{C}^n</math>. The facets are generalized (''n''β1)-cube and the [[vertex figure]] are regular [[simplex]]es. The [[regular polygon]] perimeter seen in these orthogonal projections is called a [[Petrie polygon]]. The generalized squares (''n'' = 2) are shown with edges outlined as red and blue alternating color ''p''-edges, while the higher ''n''-cubes are drawn with black outlined ''p''-edges. The number of ''m''-face elements in a ''p''-generalized ''n''-cube are: <math>p^{n-m}{n \choose m}</math>. This is ''p''<sup>''n''</sup> vertices and ''pn'' facets.<ref>{{citation | last = Coxeter | first = H. S. M. | mr = 0370328 | page = 180 | publisher = [[Cambridge University Press]] | location = London & New York | title = Regular complex polytopes | year = 1974}}.</ref> {| class=wikitable |+ Generalized hypercubes ! || ''p''=2 || ||''p''=3 ||''p''=4||''p''=5||''p''=6||''p''=7||''p''=8 |- align=center valign=bottom !valign=middle|<math>\mathbb{R}^2</math> |[[File:2-generalized-2-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|2|2}} = [[square|{4}]] = {{CDD|node_1|4|node}}<BR>4 vertices !valign=middle|<math>\mathbb{C}^2</math> |[[File:3-generalized-2-cube skew.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|3|2}} = {{CDD|3node_1|4|node}}<BR>9 vertices |[[File:4-generalized-2-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|4|2}} = {{CDD|4node_1|4|node}}<BR>16 vertices |[[File:5-generalized-2-cube skew.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|5|2}} = {{CDD|5node_1|4|node}}<BR>25 vertices |[[File:6-generalized-2-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|6|2}} = {{CDD|6node_1|4|node}}<BR>36 vertices |[[File:7-generalized-2-cube skew.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|7|2}} = {{CDD|7node_1|4|node}}<BR>49 vertices |[[File:8-generalized-2-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|8|2}} = {{CDD|8node_1|4|node}}<BR>64 vertices |- align=center valign=bottom !valign=middle|<math>\mathbb{R}^3</math> |[[File:2-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|2|3}} = [[cube|{4,3}]] = {{CDD|node_1|4|node|3|node}}<BR>8 vertices !valign=middle|<math>\mathbb{C}^3</math> |[[File:3-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|3|3}} = {{CDD|3node_1|4|node|3|node}}<BR>27 vertices |[[File:4-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|4|3}} = {{CDD|4node_1|4|node|3|node}}<BR>64 vertices |[[File:5-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|5|3}} = {{CDD|5node_1|4|node|3|node}}<BR>125 vertices |[[File:6-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|6|3}} = {{CDD|6node_1|4|node|3|node}}<BR>216 vertices |[[File:7-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|7|3}} = {{CDD|7node_1|4|node|3|node}}<BR>343 vertices |[[File:8-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|8|3}} = {{CDD|8node_1|4|node|3|node}}<BR>512 vertices |- align=center valign=bottom !valign=middle|<math>\mathbb{R}^4</math> |[[File:2-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|2|4}} = [[tesseract|{4,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node}}<BR>16 vertices !valign=middle|<math>\mathbb{C}^4</math> |[[File:3-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|3|4}} = {{CDD|3node_1|4|node|3|node|3|node}}<BR>81 vertices |[[File:4-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|4|4}} = {{CDD|4node_1|4|node|3|node|3|node}}<BR>256 vertices |[[File:5-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|5|4}} = {{CDD|5node_1|4|node|3|node|3|node}}<BR>625 vertices |[[File:6-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|6|4}} = {{CDD|6node_1|4|node|3|node|3|node}}<BR>1296 vertices |[[File:7-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|7|4}} = {{CDD|7node_1|4|node|3|node|3|node}}<BR>2401 vertices |[[File:8-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|8|4}} = {{CDD|8node_1|4|node|3|node|3|node}}<BR>4096 vertices |- align=center valign=bottom !valign=middle|<math>\mathbb{R}^5</math> |[[File:2-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|2|5}} = [[5-cube|{4,3,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node|3|node}}<BR>32 vertices !valign=middle|<math>\mathbb{C}^5</math> |[[File:3-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|3|5}} = {{CDD|3node_1|4|node|3|node|3|node|3|node}}<BR>243 vertices |[[File:4-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|4|5}} = {{CDD|4node_1|4|node|3|node|3|node|3|node}}<BR>1024 vertices |[[File:5-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|5|5}} = {{CDD|5node_1|4|node|3|node|3|node|3|node}}<BR>3125 vertices |[[File:6-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|6|5}} = {{CDD|6node_1|4|node|3|node|3|node|3|node}}<BR>7776 vertices |γ{{supsub|7|5}} = {{CDD|7node_1|4|node|3|node|3|node|3|node}}<BR>16,807 vertices |γ{{supsub|8|5}} = {{CDD|8node_1|4|node|3|node|3|node|3|node}}<BR>32,768 vertices |- align=center valign=bottom !valign=middle|<math>\mathbb{R}^6</math> |[[File:2-generalized-6-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|2|6}} = [[6-cube|{4,3,3,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}<BR>64 vertices !valign=middle|<math>\mathbb{C}^6</math> |[[File:3-generalized-6-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|3|6}} = {{CDD|3node_1|4|node|3|node|3|node|3|node|3|node}}<BR>729 vertices |[[File:4-generalized-6-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|4|6}} = {{CDD|4node_1|4|node|3|node|3|node|3|node|3|node}}<BR>4096 vertices |[[File:5-generalized-6-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|5|6}} = {{CDD|5node_1|4|node|3|node|3|node|3|node|3|node}}<BR>15,625 vertices |γ{{supsub|6|6}} = {{CDD|6node_1|4|node|3|node|3|node|3|node|3|node}}<BR>46,656 vertices |γ{{supsub|7|6}} = {{CDD|7node_1|4|node|3|node|3|node|3|node|3|node}}<BR>117,649 vertices |γ{{supsub|8|6}} = {{CDD|8node_1|4|node|3|node|3|node|3|node|3|node}}<BR>262,144 vertices |- align=center valign=bottom !valign=middle|<math>\mathbb{R}^7</math> |[[File:2-generalized-7-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|2|7}} = [[7-cube|{4,3,3,3,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<br>128 vertices !valign=middle|<math>\mathbb{C}^7</math> |[[File:3-generalized-7-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|3|7}} = {{CDD|3node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>2187 vertices |γ{{supsub|4|7}} = {{CDD|4node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>16,384 vertices |γ{{supsub|5|7}} = {{CDD|5node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>78,125 vertices |γ{{supsub|6|7}} = {{CDD|6node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>279,936 vertices |γ{{supsub|7|7}} = {{CDD|7node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>823,543 vertices |γ{{supsub|8|7}} = {{CDD|8node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>2,097,152 vertices |- align=center valign=bottom !valign=middle|<math>\mathbb{R}^8</math> |[[File:2-generalized-8-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|2|8}} = [[8-cube|{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}}<BR>256 vertices !valign=middle|<math>\mathbb{C}^8</math> |[[File:3-generalized-8-cube.svg|class=skin-invert-image|100px]]<BR>γ{{supsub|3|8}} = {{CDD|3node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>6561 vertices |γ{{supsub|4|8}} = {{CDD|4node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>65,536 vertices |γ{{supsub|5|8}} = {{CDD|5node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>390,625 vertices |γ{{supsub|6|8}} = {{CDD|6node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>1,679,616 vertices |γ{{supsub|7|8}} = {{CDD|7node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>5,764,801 vertices |γ{{supsub|8|8}} = {{CDD|8node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>16,777,216 vertices |}
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