Demihypercube


Demihypercube
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 of an n-hypercube, labeled as n 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 polytopes.

Contents

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 k21 polytope family.

The demihypercubes can be represented by extended Schläfli symbols of the form h{4,3,...,3} as half the vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes.

Constructions

They are represented by Coxeter-Dynkin diagrams of three constructive forms:

  1. CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png...CDel node h.png (As an alternated orthotope) s{2n-1}
  2. CDel node h.pngCDel 4.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.png (As an alternated hypercube) h{4,3n-1}
  3. CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.png. (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.

n  1k1  Petrie
polygon
Schläfli symbol Coxeter-Dynkin diagrams
Cn family
Dn family
Elements Facets:
Demihypercubes &
Simplexes
Vertex figure
Vertices Edges      Faces Cells 4-faces 5-faces 6-faces 7-faces 8-faces 9-faces
2 1-1,1 Demisquare
(digon)
Complete graph K2.svg
s{21}
h{4}
{31,-1,1}
CDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node.png
2 2                  
2 edges
--
3 101 demicube
(tetrahedron)
3-demicube.svg3-demicube graph.png
s{22}
h{4,3}
{31,0,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.png
4 6 4               (6 digons)
4 triangles
Triangle
(Rectified triangle)
4 111 demitesseract
(16-cell)
4-demicube.svg4-demicube graph.png
s{23}
h{4,3,3}
{31,1,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
8 24 32 16             8 demicubes
(tetrahedra)
8 tetrahedra
Octahedron
(Rectified tetrahedron)
5 121 demipenteract
5-demicube.svg5-demicube graph.png
s{24}
h{4,33}{31,2,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16 80 160 120 26           10 16-cells
16 5-cells
Rectified 5-cell
6 131 demihexeract
6-demicube.svg6-demicube graph.png
s{25}
h{4,34}{31,3,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
32 240 640 640 252 44         12 demipenteracts
32 5-simplices
Rectified hexateron
7 141 demihepteract
7-demicube.svg7-demicube graph.png
s{26}
h{4,35}{31,4,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
64 672 2240 2800 1624 532 78       14 demihexeracts
64 6-simplices
Rectified 6-simplex
8 151 demiocteract
8-demicube.svg8-demicube graph.png
s{27}
h{4,36}{31,5,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
128 1792 7168 10752 8288 4032 1136 144     16 demihepteracts
128 7-simplices
Rectified 7-simplex
9 161 demienneract
9-demicube.svg9-demicube graph.png
s{28}
h{4,37}{31,6,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
256 4608 21504 37632 36288 23520 9888 2448 274   18 demiocteracts
256 8-simplices
Rectified 8-simplex
10 171 demidekeract
10-demicube.svg10-demicube graph.png
s{29}
h{4,38}{31,7,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
512 11520 61440 122880 142464 115584 64800 24000 5300 532 20 demienneracts
512 9-simplices
Rectified 9-simplex
...
n 1n-3,1 n-demicube s{2n-1}
h{4,3n-2}{31,n-3,1}
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png...CDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png...CDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png...CDel 3.pngCDel node.png
2n-1   n (n-1)-demicubes
2n (n-1)-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 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 Dn, [3n-3,1,1] has order 2n − 1n!, and is an index 2 subgroup of the hyperoctahedral group (which is the Coxeter group BCn [4,3n-1]).

Orthotopic constructions

The rhombic disphenoid inside of a cuboid

Constructions as alternated orthotopes 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

External links


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