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.[1]

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.

The vertices and edges of a demihypercube form two copies of 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  Petrie
polygon
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 = 2nm n!/(m!(nm)!))

• 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 = 2/3 n(n−1)(n−2) 2n−3 (All original faces lost, each cube creates 4 new triangular faces)
• Cells: Dn,3 = Cn,3 + 23 Cn,4 (tetrahedra from original cells plus new ones)
• Hypercells: Dn,4 = Cn,4 + 24 Cn,5 (16-cells and 5-cells respectively)
• ...
• [For m = 3,...,n−1]: Dn,m = Cn,m + 2m Cn,m+1 (m-demicubes and m-simplexes respectively)
• ...
• Facets: Dn,n−1 = 2n + 2n−1 ((n−1)-demicubes and (n−1)-simplices respectively)

## Symmetry group

The stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group ${\displaystyle BC_{n))$ [4,3n−1]) has index 2. It is the Coxeter group ${\displaystyle D_{n},}$ [3n−3,1,1] of order ${\displaystyle 2^{n-1}n!}$, and is generated by permutations of the coordinate axes and reflections along pairs of coordinate axes.[2]

## Orthotopic constructions

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.

## References

1. ^ Regular and semi-regular polytopes III, p. 315-316
2. ^ "week187". math.ucr.edu. Retrieved 20 April 2018.
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• 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)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds