141.35.40.137: K needs to be alg.closed for the uniqueness

2013-07-02T18:17:52Z

K needs to be alg.closed for the uniqueness

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{{distinguish|Hemicube (geometry)}}
[[File:CubeAndStel.svg|thumb|[[Alternation (geometry)|Alternation]] of the ''n''-cube yields one of two ''n''-demicubes, as in this 3-dimensional illustration of the two [[tetrahedra]] that arise as the 3-demicubes of the 3-cube.]]
In [[geometry]], '''demihypercubes''' (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of n-[[polytopes]] constructed from [[Alternation (geometry)|alternation]] of an n-[[hypercube]], labeled as ''hγn'' for being ''half'' of the hypercube family, ''γn''. Half of the vertices are deleted and new facets are formed. The ''2n'' facets become ''2n'' '''(n-1)-demicubes''', and 2n '''(n-1)-simplex''' facets are formed in place of the deleted vertices.

They have been named with a ''demi-'' prefix to each [[hypercube]] name: demicube, demitesseract, etc. The demicube is identical to the regular [[tetrahedron]], and the demitesseract is identical to the regular [[16-cell]]. The [[demipenteract]] is considered ''semiregular'' for having only regular facets. Higher forms don't have all regular facets but are all [[uniform polytope]]s.

The vertices and edges of a demihypercube form two copies of the [[halved cube graph]].

== Discovery ==

[[Thorold Gosset]] described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in n-dimensions above 3. He called it a ''5-ic semi-regular''. It also exists within the [[Semiregular k 21 polytope|semiregular k21 polytope]] family.

The demihypercubes can be represented by extended [[Schläfli symbol]]s of the form h{4,3,...,3} as half the vertices of {4,3,...,3}. The [[vertex figure]]s of demihypercubes are [[Rectification (geometry)|rectified]] n-[[simplex]]es.

== Constructions ==

They are represented by [[Coxeter-Dynkin diagram]]s of three constructive forms:
#{{CDD|node_h|2c|node_h|2c|node_h}}...{{CDD|node_h}} (As an [[Alternation (geometry)|alternated]] [[orthotope]]) sr{2n-1}
#{{CDD|node_h|4|node|3}}...{{CDD|3|node}} (As an alternated [[hypercube]]) h{4,3n-1}
#{{CDD|nodes_10ru|split2|node|3}}...{{CDD|3|node}}. (As a demihypercube) {31,n-3,1}

[[H.S.M. Coxeter]] also labeled the third bifurcating diagrams as '''1k1''' representing the lengths of the 3 branches and lead by the ringed branch.

An ''n-demicube'', ''n'' greater than 2, has ''n*(n-1)/2'' edges meeting at each vertex. The graphs below show less edges at each vertex due to overlapping edges in the symmetry projection.

{| class="wikitable"
!rowspan=2|''n''
!rowspan=2| 1k1 
!rowspan=2|[[Petrie polygon|Petrie polygon]]
!rowspan=2|[[Schläfli symbol]]
!rowspan=2|[[Coxeter-Dynkin diagram]]s A1n family BCn family Dn family
!colspan=10|Elements
!rowspan=2|[[Facet (geometry)|Facets]]: Demihypercubes & Simplexes
!rowspan=2|[[Vertex figure]]
|-
!Vertices
!Edges     
!Faces
!Cells
!4-faces
!5-faces
!6-faces
!7-faces
!8-faces
!9-faces
|-
![[2-polytope|2]]
! 1-1,1
|align=center|''demisquare'' ([[digon]]) [[Image:Complete graph K2.svg|60px]]
|sr{21} h{4} {31,-1,1}
|width=150|{{CDD|node_h|2c|node_h}} {{CDD|node_h|4|node}} {{CDD|node_1|2c|node}}
|2
|2
| 
| 
| 
| 
| 
| 
| 
| 
|  2 [[Edge (geometry)|edge]]s
| --
|-
![[3-polytope|3]]
! 101
|align=center|''demicube'' ([[tetrahedron]]) [[File:3-demicube.svg|60px]][[File:3-demicube_t0_B3.svg|60px]]
|sr{22} h{4,3} {31,0,1}
|{{CDD|node_h|2c|node_h|2c|node_h}} {{CDD|node_h|4|node|3|node}} {{CDD|nodes_10ru|split2|node}}
|4
|6
|4
| 
| 
| 
| 
| 
| 
| 
| ''(6 [[digon]]s)'' 4 [[triangle]]s
|Triangle (Rectified triangle)
|-
![[4-polytope|4]]
! 111
|align=center|[[demitesseract]] ([[16-cell]]) [[File:4-demicube_t0_D4.svg|60px]][[File:4-demicube_t0_B4.svg|60px]]
|sr{23} h{4,3,3} {31,1,1}
|{{CDD|node_h|2c|node_h|2c|node_h|2c|node_h}} {{CDD|node_h|4|node|3|node|3|node}} {{CDD|nodes_10ru|split2|node|3|node}}
|8
|24
|32
|16
| 
| 
| 
| 
| 
| 
|8 demicubes (tetrahedra) 8 [[tetrahedron|tetrahedra]]
|[[Octahedron]] (Rectified tetrahedron)
|-
![[5-polytope|5]]
! 121
|align=center|[[demipenteract]] [[File:5-demicube_t0_D5.svg|60px]][[File:5-demicube_t0_B5.svg|60px]]
|sr{24} h{4,33}{31,2,1}
|{{CDD|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h}} {{CDD|node_h|4|node|3|node|3|node|3|node}} {{CDD|nodes_10ru|split2|node|3|node|3|node}}
|16
|80
|160
|120
|26
| 
| 
| 
| 
| 
|10 [[16-cell]]s 16 [[5-cell]]s
|[[Rectified 5-cell]]
|-
![[6-polytope|6]]
! 131
|align=center|[[demihexeract]] [[File:6-demicube_t0_D6.svg|60px]][[File:6-demicube_t0_B6.svg|60px]]
|sr{25} h{4,34}{31,3,1}
|{{CDD|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h}} {{CDD|node_h|4|node|3|node|3|node|3|node|3|node}} {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node}}
|32
|240
|640
|640
|252
|44
| 
| 
| 
| 
|12 ''demipenteracts'' 32 5-[[Hexateron|simplices]]
|[[Rectified hexateron]]
|-
![[7-polytope|7]]
! 141
|align=center|[[demihepteract]] [[File:7-demicube_t0_D7.svg|60px]][[File:7-demicube_t0_B7.svg|60px]]
|sr{26} h{4,35}{31,4,1}
|{{CDD|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h}} {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}}
|64
|672
|2240
|2800
|1624
|532
|78
| 
| 
| 
|14 ''demihexeracts'' 64 6-[[Simplex|simplices]]
|[[Rectified 6-simplex]]
|-
![[8-polytope|8]]
! 151
|align=center|[[demiocteract]] [[File:8-demicube_t0_D8.svg|60px]][[File:8-demicube_t0_B8.svg|60px]]
|sr{27} h{4,36}{31,5,1}
|{{CDD|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h}} {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node}}
|128
|1792
|7168
|10752
|8288
|4032
|1136
|144
| 
| 
|16 ''demihepteracts'' 128 7-[[Simplex|simplices]]
|[[Rectified 7-simplex]]
|-
![[9-polytope|9]]
! 161
|align=center|[[demienneract]] [[File:9-demicube_t0_D9.svg|60px]][[File:9-demicube_t0_B9.svg|60px]]
|sr{28} h{4,37}{31,6,1}
|{{CDD|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h}} {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|256
|4608
|21504
|37632
|36288
|23520
|9888
|2448
|274
| 
|18 ''demiocteracts'' 256 8-[[Simplex|simplices]]
|[[Rectified 8-simplex]]
|-
![[10-polytope|10]]
! 171
|align=center|[[demidekeract]] [[File:10-demicube.svg|60px]][[File:10-demicube graph.png|60px]]
|sr{29} h{4,38}{31,7,1}
|{{CDD|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h}} {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|512
|11520
|61440
|122880
|142464
|115584
|64800
|24000
|5300
|532
|20 ''demienneracts'' 512 9-[[Simplex|simplices]]
|[[Rectified 9-simplex]]
|-
|...
|-
![[polytope|n]]
! 1n-3,1
|align=center|''n-demicube''
|sr{2n-1} h{4,3n-2}{31,n-3,1}
|{{CDD|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h|2c|node_h}}...{{CDD|node_h}} {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node}}...{{CDD|3|node}} {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}}...{{CDD|3|node}}
|2n-1
|colspan=9| 
|n (n-1)-demicubes 2n (n-1)-[[Simplex|simplices]]
|Rectified (n-1)-simplex
|}

In general, a demicube's elements can be determined from the original n-cube: (With Cn,m = ''mth''-face count in [[hypercube|n-cube]] = 2n-m*n!/(m!*(n-m)!))
* '''Vertices:''' Dn,0 = 1/2 * Cn,0 = 2n-1 (Half the n-cube vertices remain)
* '''Edges:''' Dn,1 = Cn,2 = 1/2 n(n-1)2n-2 (All original edges lost, each square faces create a new edge)
* '''Faces:''' Dn,2 = 4 * Cn,3 = n(n-1)(n-2)2n-3 (All original faces lost, each cube creates 4 new triangular faces)
* '''Cells:''' Dn,3 = Cn,3 + 2n-4Cn,4 (tetrahedra from original cells plus new ones)
* '''Hypercells:''' Dn,4 = Cn,4 + 2n-5Cn,5 (16-cells and 5-cells respectively)
* ...
* [For m=3...n-1]: Dn,m = Cn,m + 2n-1-mCn,m+1 (m-demicubes and m-simplexes respectively)
*...
* '''Facets:''' Dn,n-1 = n + 2n ((n-1)-demicubes and (n-1)-simplices respectively)

== Symmetry group ==
The symmetry group of the demihypercube is the [[Coxeter group]] <math>D_n,</math> [3n-3,1,1] has order <math>2^{n-1}n!,</math> and is an index 2 subgroup of the [[hyperoctahedral group]] (which is the Coxeter group <math>BC_n</math> [4,3n-1]).

== Orthotopic constructions ==
[[File:Rhombic disphenoid.png|thumb|The rhombic disphenoid inside of a [[cuboid]]]]

Constructions as alternated [[orthotope]]s have the same topology, but can be stretched with different lengths in ''n''-axes of symmetry.

The [[rhombic disphenoid]] is the three-dimensional example as alternated cuboid. It has three sets of edge lengths, and [[scalene triangle]] faces.

== See also ==
* [[Hypercube honeycomb]]
* [[Semiregular E-polytope]]

== References ==
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)

== External links ==
*{{GlossaryForHyperspace | anchor=half | title=Half measure polytope }}

{{Dimension topics}}
{{Polytopes}}

[[Category:Multi-dimensional geometry]]
[[Category:Polytopes]]

Root datum - Revision history

141.35.40.137: K needs to be alg.closed for the uniqueness