View source for 7-demicube

{{Short description|Uniform 7-polytope}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=3|Demihepteract<BR>(7-demicube)
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Demihepteract ortho petrie.svg|280px]]<BR>[[Petrie polygon]] projection
|-
|bgcolor=#e7dcc3|Type
|colspan=2|Uniform [[7-polytope]]
|-
|bgcolor=#e7dcc3|Family
|colspan=2|[[demihypercube]]
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]
|colspan=2| 1<sub>41</sub>
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|{3,3<sup>4,1</sup>} = h{4,3<sup>5</sup>}<BR>s{2<sup>1,1,1,1,1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s
|colspan=2|{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>
{{CDD|node_h|2x|node_h|4|node|3|node|3|node|3|node|3|node|}}<BR>
{{CDD|node_h|2x|node_h|2x|node_h|4|node|3|node|3|node|3|node|}}<BR>
{{CDD|node_h|2x|node_h|2x|node_h|2x|node_h|4|node|3|node|3|node|}}<BR>
{{CDD|node_h|2x|node_h|2x|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|2x|node_h|2x|node_h|4|node|}}<BR>
{{CDD|node_h|2x|node_h|2x|node_h|2x|node_h|2x|node_h|2x|node_h|2x|node_h}}
|-
|bgcolor=#e7dcc3|6-faces||78||14 [[demihexeract|{3<sup>1,3,1</sup>}]][[Image:Demihexeract ortho petrie.svg|35px]]<BR>64 [[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||532||84 [[demipenteract|{3<sup>1,2,1</sup>}]][[Image:Demipenteract graph ortho.svg|25px]]<BR>448 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||1624||280 [[16-cell|{3<sup>1,1,1</sup>}]][[Image:4-orthoplex.svg|25px]]<BR>1344 [[5-cell|{3<sup>3</sup>}]][[Image:4-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||2800||560 [[Tetrahedron|{3<sup>1,0,1</sup>}]][[Image:3-simplex_t0.svg|25px]]<BR>2240 [[Tetrahedron|{3,3}]][[Image:3-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||2240||[[Triangle|{3}]][[Image:2-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||colspan=2|672
|-
|bgcolor=#e7dcc3|Vertices||colspan=2|64
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Rectified 6-simplex]]<BR>[[File:6-simplex t1.svg|40px]]
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry group]]
|colspan=2|D<sub>7</sub>, [3<sup>4,1,1</sup>] = [1<sup>+</sup>,4,3<sup>5</sup>]<BR>[2<sup>6</sup>]<sup>+</sup>
|-
|bgcolor=#e7dcc3|Dual
|colspan=2|?
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|}
In [[geometry]], a '''demihepteract''' or '''7-demicube''' is a uniform [[7-polytope]], constructed from the 7-hypercube ([[hepteract]]) with [[Alternation (geometry)|alternated]] vertices removed. It is part of a dimensionally infinite family of [[uniform polytope]]s called [[demihypercube]]s.

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as HM<sub>7</sub> for a 7-dimensional ''half measure'' polytope.

[[Coxeter]] named this polytope as '''1<sub>41</sub>''' from its [[Coxeter diagram]], with a ring on one of the 1-length branches, {{CDD|node_1|3|node|split1|nodes|3a|nodea|3a|nodea|3a|nodea}} and [[Schläfli symbol]] <math>\left\{3 \begin{array}{l}3, 3, 3, 3\\3\end{array}\right\}</math> or {3,3<sup>4,1</sup>}.

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a demihepteract centered at the origin are alternate halves of the [[hepteract]]:
: (±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.

== Images ==
{{7-demicube Coxeter plane graphs|t0|125}}

== As a configuration==
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 7-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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/hax.htm|x3o3o *b3o3o3o - hax}}</ref>
{| class=wikitable
!D<sub>7</sub>||{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}||[[Coxeter symbol|''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>||colspan=2|f<sub>5</sub>||colspan=2|f<sub>6</sub>|| [[vertex figure|''k''-figures]]|| notes
|- align=right
|A<sub>6</sub> ||{{CDD|nodea_x|2|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}|| ( )
! f<sub>0</sub> 
|BGCOLOR="#ffe0ff"|'''64'''||21||105||35||140||35||105||21||42||7||7||[[Rectified 6-simplex|0<sub>41</sub>]] || D<sub>7</sub>/A<sub>6</sub> = 64*7!/7! = 64
|- align=right
|A<sub>4</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|nodea_1|2|nodes_x0|2|nodea|3a|nodea|3a|nodea|3a|nodea}}|| { }
! f<sub>1</sub> 
||2||BGCOLOR="#ffe0e0"|'''672'''||10||5||20||10||20||10||10||5||2||[[5-cell prism|{ }×{3,3,3}]] || D<sub>7</sub>/A<sub>4</sub>A<sub>1</sub>A<sub>1</sub> = 64*7!/5!/2/2 = 672
|- align=right
|A<sub>3</sub>A<sub>2</sub> ||{{CDD|nodea_1|3a|nodes_0x|2|nodea_x|2|nodea|3a|nodea|3a|nodea}}|| [[triangle|1<sub>00</sub>]]
! f<sub>2</sub> 
||3||3||BGCOLOR="#ffffe0"|'''2240'''||1||4||4||6||6||4||4||1||[[Tetrahedral pyramid|{3,3}v( )]] || D<sub>7</sub>/A<sub>3</sub>A<sub>2</sub> = 64*7!/4!/3! = 2240
|- align=right
|A<sub>3</sub>A<sub>3</sub> ||{{CDD|nodea_1|3a|branch|2|nodea_x|2|nodea|3a|nodea|3a|nodea}}|| [[tetrahedron|1<sub>01</sub>]]
!rowspan=2| f<sub>3</sub> 
||4||6||4||BGCOLOR="#e0ffe0"|'''560'''||BGCOLOR="#e0ffe0"|*||4||0||6||0||4||0||[[Tetrahedron|{3,3}]] || D<sub>7</sub>/A<sub>3</sub>A<sub>3</sub> = 64*7!/4!/4! = 560
|- align=right
|A<sub>3</sub>A<sub>2</sub> ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|2|nodea_x|2|nodea|3a|nodea}} || [[tetrahedron|1<sub>10</sub>]]
||4||6||4||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''2240'''||1||3||3||3||3||1||[[Triangular pyramid|{3}v( )]] || D<sub>7</sub>/A<sub>3</sub>A<sub>2</sub> = 64*7!/4!/3! = 2240
|- align=right
|D<sub>4</sub>A<sub>2</sub> ||{{CDD|nodea_1|3a|branch|3a|nodea|2|nodea_x|2|nodea|3a|nodea}}|| [[16-cell|1<sub>11</sub>]]
!rowspan=2| f<sub>4</sub> 
||8||24||32||8||8||BGCOLOR="#e0ffff"|'''280'''||BGCOLOR="#e0ffff"|*||3||0||3||0||[[Triangle|{3}]] || D<sub>7</sub>/D<sub>4</sub>A<sub>2</sub> = 64*7!/8/4!/2 = 280
|- align=right
|A<sub>4</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|3a|nodea|2|nodea_x|2|nodea}}|| [[5-cell|1<sub>20</sub>]]
||5||10||10||0||5||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''1344'''||1||2||2||1||[[Isosceles triangle|{ }v( )]] || D<sub>7</sub>/A<sub>4</sub>A<sub>1</sub> = 64*7!/5!/2 = 1344
|- align=right
|D<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|2|nodea_x|2|nodea}}|| [[5-demicube|1<sub>21</sub>]]
!rowspan=2| f<sub>5</sub> 
||16||80||160||40||80||10||16||BGCOLOR="#e0e0ff"|'''84'''||BGCOLOR="#e0e0ff"|*||2||0||rowspan=2|{ } || D<sub>7</sub>/D<sub>5</sub>A<sub>1</sub> = 64*7!/16/5!/2 = 84
|- align=right
|A<sub>5</sub> ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|3a|nodea|3a|nodea|2|nodea_x}}|| [[5-simplex|1<sub>30</sub>]]
||6||15||20||0||15||0||6||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''448'''||1||1|| D<sub>7</sub>/A<sub>5</sub> = 64*7!/6! = 448
|- align=right
|D<sub>6</sub> ||{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|2|nodea_x}}|| [[6-demicube|1<sub>31</sub>]]
!rowspan=2| f<sub>6</sub> 
||32||240||640||160||480||60||192||12||32||BGCOLOR="#ffe0ff"|'''14'''||BGCOLOR="#ffe0ff"|*||rowspan=2|( ) || D<sub>7</sub>/D<sub>6</sub> = 64*7!/32/6! = 14
|- align=right
|A<sub>6</sub> ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}|| [[6-simplex|1<sub>40</sub>]]
||7||21||35||0||35||0||21||0||7||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|'''64'''|| D<sub>7</sub>/A<sub>6</sub> = 64*7!/7! = 64
|}

== Related polytopes ==

There are 95 uniform polytopes with D<sub>6</sub> symmetry, 63 are shared by the B<sub>6</sub> symmetry, and 32 are unique:
{{Demihepteract_family}}

== References ==
{{Reflist}}
* [[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|polyexa.htm|7D uniform polytopes (polyexa)|x3o3o *b3o3o3o3o - hesa}}

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

{{Polytopes}}

[[Category:7-polytopes]]