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 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 - 1 (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 do not have all regular facets but are all uniform polytopes.

The vertex-edge graph of the demihypercube is the halved cube graph.

An n-demicube has inversion symmetry if n is even.

Discovery

Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in n-dimensions above three. 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. ... (As an alternated orthotope) s{21,1,...,1}
  2. ... (As an alternated hypercube) h{4,3n−1}
  3. .... (As a demihypercube) {31,n−3,1}

H.S.M. Coxeter also labeled the third bifurcating diagrams as 1k1 representing the lengths of the three branches and led 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  Coxeter plane
projection		 Schläfli symbol Coxeter diagrams
A1n
Bn
Dn		 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)
 s{2}
h{4}
{31,−1,1}

2 2                  
2 edges		 --
3 101 demicube
(tetrahedron)
 s{21,1}
h{4,3}
{31,0,1}

4 6 4               (6 digons)
4 triangles		 Triangle
(Rectified triangle)
4 111 demitesseract
(16-cell)
 s{21,1,1}
h{4,3,3}
{31,1,1}

8 24 32 16             8 demicubes
(tetrahedra)
8 tetrahedra		 Octahedron
(Rectified tetrahedron)
5 121 demipenteract
 s{21,1,1,1}
h{4,33}{31,2,1}

16 80 160 120 26           10 16-cells
16 5-cells		 Rectified 5-cell
6 131 demihexeract
 s{21,1,1,1,1}
h{4,34}{31,3,1}

32 240 640 640 252 44         12 demipenteracts
32 5-simplices		 Rectified hexateron
7 141 demihepteract
 s{21,1,1,1,1,1}
h{4,35}{31,4,1}

64 672 2240 2800 1624 532 78       14 demihexeracts
64 6-simplices		 Rectified 6-simplex
8 151 demiocteract
 s{21,1,1,1,1,1,1}
h{4,36}{31,5,1}

128 1792 7168 10752 8288 4032 1136 144     16 demihepteracts
128 7-simplices		 Rectified 7-simplex
9 161 demienneract
 s{21,1,1,1,1,1,1,1}
h{4,37}{31,6,1}

256 4608 21504 37632 36288 23520 9888 2448 274   18 demiocteracts
256 8-simplices		 Rectified 8-simplex
10 171 demidekeract
 s{21,1,1,1,1,1,1,1,1}
h{4,38}{31,7,1}

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{21,1,...,1}
h{4,3n−2}{31,n−3,1}		 ...
...
...		 2n−1   2n (n − 1)-demicubes
2n−1 (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)!))

Symmetry group

The stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group [4,3n−1]) has index 2. It is the Coxeter group [3n−3,1,1] of order , and is generated by permutations of the coordinate axes and reflections along pairs of coordinate axes.

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

References

  1. ↑ Regular and semi-regular polytopes III, p. 315–316
  2. ↑ "week187". math.ucr.edu. Retrieved 20 April 2018.
Fundamental convex regular and uniform polytopes
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations