View source for 5-demicube

{{Short description|Regular 5-polytope}}
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=3|Demipenteract<BR>(5-demicube)
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Demipenteract graph ortho.svg|300px]]<BR>[[Petrie polygon]] projection
|-
|bgcolor=#e7dcc3 width=80|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|Family (D<sub>n</sub>)
|colspan=2|5-[[demihypercube|demicube]]
|-
|bgcolor=#e7dcc3|Families (E<sub>n</sub>)
|colspan=2|[[Semiregular k 21 polytope|k<sub>21</sub> polytope]]<BR>[[Uniform 1 k2 polytope|1<sub>k2</sub> polytope]]
|-
|bgcolor=#e7dcc3|[[Coxeter symbol|Coxeter<BR>symbol]]
|colspan=2| 1<sub>21</sub>
|-
|bgcolor=#e7dcc3|[[Schläfli symbol|Schläfli<BR>symbols]]
|colspan=2|{3,3<sup>2,1</sup>} = h{4,3<sup>3</sup>}<BR>s{2,4,3,3} or h{2}h{4,3,3}<BR>sr{2,2,4,3} or h{2}h{2}h{4,3}<BR>h{2}h{2}h{2}h{4}<BR>s{2<sup>1,1,1,1</sup>} or h{2}h{2}h{2}s{2}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram|Coxeter<BR>diagrams]]
|colspan=2|{{CDD|nodes_10ru|split2|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node}}<BR>{{CDD|node_h|2x|node_h|4|node|3|node|3|node|}}<BR>{{CDD|node_h|2x|node_h|2x|node_h|4|node|3|node|}}<BR>{{CDD|node_h|2x|node_h|2x|node_h|2x|node_h|4|node|}}<BR>{{CDD|node_h|2x|node_h|2x|node_h|2x|node_h|2x|node_h|}}
|-
|bgcolor=#e7dcc3|4-faces||26||10 [[16-cell|{3<sup>1,1,1</sup>}]][[Image:Cross graph 4.svg|25px]]<BR>16 [[5-cell|{3,3,3}]][[Image:4-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||120||40 [[Tetrahedron|{3<sup>1,0,1</sup>}]][[Image:3-simplex t0.svg|25px]]<BR>80 [[Tetrahedron|{3,3}]][[Image:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||160||[[Triangle|{3}]][[Image:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||colspan=2|80
|-
|bgcolor=#e7dcc3|Vertices||colspan=2|16
|-
|bgcolor=#e7dcc3|[[Vertex figure|Vertex<BR>figure]]
|colspan=2|[[File:5-demicube verf.svg|60px]]<BR>[[rectified 5-cell]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon|Petrie<BR>polygon]]
|colspan=2|[[Octagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]
|colspan=2|D<sub>5</sub>, [3<sup>2,1,1</sup>] = [1<sup>+</sup>,4,3<sup>3</sup>]<BR>[2<sup>4</sup>]<sup>+</sup>
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|}
In [[Five-dimensional space|five-dimensional]] [[geometry]], a '''demipenteract''' or '''5-demicube''' is a semiregular [[5-polytope]], constructed from a ''5-hypercube'' ([[penteract]]) with [[Alternation (geometry)|alternated]] vertices removed.

It was discovered by [[Thorold Gosset]]. Since it was the only [[Semiregular polytope|semiregular]] 5-polytope (made of more than one type of regular [[Facet (geometry)|facets]]), he called it a [[Semiregular polytope|5-ic semi-regular]]. [[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as HM<sub>5</sub> for a 5-dimensional ''half measure'' polytope.

[[Coxeter]] named this polytope as '''1<sub>21</sub>''' from its [[Coxeter diagram]], which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, {{CDD|node_1|3|node|split1|nodes|3a|nodea}} and [[Schläfli symbol]] <math>\left\{3 \begin{array}{l}3, 3\\3\end{array}\right\}</math> or {3,3<sup>2,1</sup>}.

It exists in the [[semiregular k 21 polytope|k<sub>21</sub> polytope]] family as 1<sub>21</sub> with the Gosset polytopes: [[Gosset 2 21 polytope|2<sub>21</sub>]], [[Gosset 3 21 polytope|3<sub>21</sub>]], and [[Gosset 4 21 polytope|4<sub>21</sub>]].

The graph formed by the vertices and edges of the demipenteract is sometimes called the [[Clebsch graph]], though that name sometimes refers to the [[folded cube graph]] of order five instead.

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a demipenteract centered at the origin and edge length 2{{radic|2}} are alternate halves of the [[penteract]]:
: (±1,±1,±1,±1,±1)
with an odd number of plus signs.

== As a configuration ==
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 5-demicube. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>

The diagonal f-vector numbers are derived through the [[Wythoff construction]], dividing the full group order of a subgroup order by removing one mirror at a time.<ref>{{KlitzingPolytopes|../incmats/hin.htm| x3o3o *b3o3o - hin}}</ref>

{| class=wikitable
!D<sub>5</sub>||{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea}}||k-face|| f<sub>k</sub> || f<sub>0</sub> || f<sub>1</sub>||f<sub>2</sub>||colspan=2|f<sub>3</sub>||colspan=2|f<sub>4</sub>|| ''k''-figure|| notes(*)
|- align=right
|A<sub>4</sub> ||{{CDD|nodea_x|2|branch|3a|nodea|3a|nodea}}||(&nbsp;)
! f<sub>0</sub>
|BGCOLOR="#ffe0e0"|'''16'''||10||30||10||20||5||5||[[rectified 5-cell]] || D<sub>5</sub>/A<sub>4</sub> = 16*5!/5! = 16
|- align=right
|A<sub>2</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|nodea_1|2|nodes_x0|2|nodea|3a|nodea}}||{&nbsp;}
! f<sub>1</sub>
|2||BGCOLOR="#ffffe0"|'''80'''||6||3||6||3||2||[[triangular prism]] || D<sub>5</sub>/A<sub>2</sub>A<sub>1</sub>A<sub>1</sub> = 16*5!/3!/2/2 = 80
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|nodes_0x|2|nodea_x|2|nodea}}||[[triangle|{3}]]
! f<sub>2</sub>
|3||3||BGCOLOR="#e0ffe0"|'''160'''||1||2||2||1||[[Isosceles triangle]] || D<sub>5</sub>/A<sub>2</sub>A<sub>1</sub> = 16*5!/3!/2 = 160
|- align=right
|A<sub>3</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|branch|2|nodea_x|2|nodea}}||[[tetrahedron|h{4,3}]]
!rowspan=2| f<sub>3</sub>
|4||6||4||BGCOLOR="#e0ffff"|'''40'''||BGCOLOR="#e0ffff"|*||2||0||Segment {&nbsp;} || D<sub>5</sub>/A<sub>3</sub>A<sub>1</sub> = 16*5!/4!/2 = 40
|- align=right
|A<sub>3</sub> ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|2|nodea_x}}||[[tetrahedron|{3,3}]]
|4||6||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''80'''||1||1||Segment {&nbsp;} || D<sub>5</sub>/A<sub>3</sub> = 16*5!/4! = 80
|- align=right
|D<sub>4</sub> ||{{CDD|nodea_1|3a|branch|3a|nodea|2|nodea_x}}||[[16-cell|h{4,3,3}]]
!rowspan=2| f<sub>4</sub>
|8||24||32||8||8||BGCOLOR="#e0e0ff"|'''10'''||BGCOLOR="#e0e0ff"|'''*'''||Point (&nbsp;) || D<sub>5</sub>/D<sub>4</sub> = 16*5!/8/4! = 10
|- align=right
|A<sub>4</sub> ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|3a|nodea}}||[[5-cell|{3,3,3}]]
|5||10||10||0||5||BGCOLOR="#e0e0ff"|'''*'''||BGCOLOR="#e0e0ff"|'''16'''||Point (&nbsp;) || D<sub>5</sub>/A<sub>4</sub>  = 16*5!/5! = 16
|}
'''*''' = The number of elements (diagonal values) can be computed by the symmetry order D<sub>5</sub> divided by the symmetry order of the subgroup with selected mirrors removed.

== Projected images ==
{| class=wikitable width=240
|[[Image:Demipenteract wf.png|240px]]<BR>[[Perspective projection]].
|}

==Images==
{{5-demicube Coxeter plane graphs|t0|200}}

== Related polytopes ==

It is a part of a dimensional family of [[uniform polytope]]s called [[demihypercube]]s for being [[Alternation (geometry)|alternation]] of the [[hypercube]] family.

There are 23 [[Uniform 5-polytope]]s (uniform 5-polytopes) that can be constructed from the D<sub>5</sub> symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the [[penteract]]ic family.

{{Demipenteract family}}

The 5-demicube is third in a dimensional series of [[Uniform k21 polytope|semiregular polytope]]s. Each progressive [[uniform polytope]] is constructed [[vertex figure]] of the previous polytope. [[Thorold Gosset]] identified this series in 1900 as containing all [[regular polytope]] facets, containing all [[simplex]]es and [[orthoplex]]es ([[5-simplex|5-simplices]] and [[5-orthoplex|5-orthoplexes]] in the case of the 5-demicube). In [[Coxeter]]'s notation the 5-demicube is given the symbol 1<sub>21</sub>.
{{Gosset_semiregular_polytopes}}

{{1 k2 polytopes}}

== References ==
<references />
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1<sub>n1</sub>)
* {{KlitzingPolytopes|polytera.htm|5D uniform polytopes (polytera)|x3o3o *b3o3o - hin}}

== External links ==
*{{GlossaryForHyperspace | anchor=half | title=Demipenteract }}
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:5-polytopes]]