# 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.

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 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. Template:CDD...Template:CDD (As an alternated orthotope) sr{2n-1}
2. Template:CDD...Template:CDD (As an alternated hypercube) h{4,3n-1}
3. Template:CDD...Template:CDD. (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
A1n family
BCn 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)
sr{21}
h{4}
{31,-1,1}
Template:CDD
Template:CDD
Template:CDD
2 2
2 edges
--
3 101 demicube
(tetrahedron)
sr{22}
h{4,3}
{31,0,1}
Template:CDD
Template:CDD
Template:CDD
4 6 4               (6 digons)
4 triangles
Triangle
(Rectified triangle)
4 111 demitesseract
(16-cell)
sr{23}
h{4,3,3}
{31,1,1}
Template:CDD
Template:CDD
Template:CDD
8 24 32 16             8 demicubes
(tetrahedra)
8 tetrahedra
Octahedron
(Rectified tetrahedron)
5 121 demipenteract
sr{24}
h{4,33}{31,2,1}
Template:CDD
Template:CDD
Template:CDD
16 80 160 120 26           10 16-cells
16 5-cells
Rectified 5-cell
6 131 demihexeract
sr{25}
h{4,34}{31,3,1}
Template:CDD
Template:CDD
Template:CDD
32 240 640 640 252 44         12 demipenteracts
32 5-simplices
Rectified hexateron
7 141 demihepteract
sr{26}
h{4,35}{31,4,1}
Template:CDD
Template:CDD
Template:CDD
64 672 2240 2800 1624 532 78       14 demihexeracts
64 6-simplices
Rectified 6-simplex
8 151 demiocteract
sr{27}
h{4,36}{31,5,1}
Template:CDD
Template:CDD
Template:CDD
128 1792 7168 10752 8288 4032 1136 144     16 demihepteracts
128 7-simplices
Rectified 7-simplex
9 161 demienneract
sr{28}
h{4,37}{31,6,1}
Template:CDD
Template:CDD
Template:CDD
256 4608 21504 37632 36288 23520 9888 2448 274   18 demiocteracts
256 8-simplices
Rectified 8-simplex
10 171 demidekeract
sr{29}
h{4,38}{31,7,1}
Template:CDD
Template:CDD
Template:CDD
512 11520 61440 122880 142464 115584 64800 24000 5300 532 20 demienneracts
512 9-simplices
Rectified 9-simplex
...
n 1n-3,1 n-demicube sr{2n-1}
h{4,3n-2}{31,n-3,1}
Template:CDD...Template:CDD
Template:CDD...Template:CDD
Template:CDD...Template:CDD
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 ${\displaystyle D_{n},}$ [3n-3,1,1] has order ${\displaystyle 2^{n-1}n!,}$ and is an index 2 subgroup of the hyperoctahedral group (which is the Coxeter group ${\displaystyle BC_{n}}$ [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.

## References

• 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)