# Demihypercube

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 *hγ _{n}* for being

*half*of the hypercube family,

*γ*. Half of the vertices are deleted and new facets are formed. The

_{n}*2n*facets become

*2n*

**(n-1)-demicubes**, and 2

^{n}

**(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 3. He called it a *5-ic semi-regular*. It also exists within the semiregular k_{21} 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:

... (As an alternated orthotope) s{2 ^{1,1...,1}}... (As an alternated hypercube) h{4,3 ^{n-1}}... . (As a demihypercube) {3 ^{1,n-3,1}}

H.S.M. Coxeter also labeled the third bifurcating diagrams as **1 _{k1}** representing the lengths of the 3 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 |
1_{k1} |
Petrie polygon |
Schläfli symbol | Coxeter diagrams A _{1}^{n}B _{n}D _{n} |
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} {3 ^{1,−1,1}} |
2 | 2 | 2 edges |
-- | |||||||||

3 | 1_{01} |
demicube(tetrahedron) |
s{2^{1,1}}h{4,3} {3 ^{1,0,1}} |
4 | 6 | 4 | (6 digons)4 triangles |
Triangle (Rectified triangle) | ||||||||

4 | 1_{11} |
demitesseract (16-cell) |
s{2^{1,1,1}}h{4,3,3} {3 ^{1,1,1}} |
8 | 24 | 32 | 16 | 8 demicubes (tetrahedra) 8 tetrahedra |
Octahedron (Rectified tetrahedron) | |||||||

5 | 1_{21} |
demipenteract |
s{2^{1,1,1,1}}h{4,3 ^{3}}{3^{1,2,1}} |
16 | 80 | 160 | 120 | 26 | 10 16-cells 16 5-cells |
Rectified 5-cell | ||||||

6 | 1_{31} |
demihexeract |
s{2^{1,1,1,1,1}}h{4,3 ^{4}}{3^{1,3,1}} |
32 | 240 | 640 | 640 | 252 | 44 | 12 demipenteracts32 5-simplices |
Rectified hexateron | |||||

7 | 1_{41} |
demihepteract |
s{2^{1,1,1,1,1,1}}h{4,3 ^{5}}{3^{1,4,1}} |
64 | 672 | 2240 | 2800 | 1624 | 532 | 78 | 14 demihexeracts64 6-simplices |
Rectified 6-simplex | ||||

8 | 1_{51} |
demiocteract |
s{2^{1,1,1,1,1,1,1}}h{4,3 ^{6}}{3^{1,5,1}} |
128 | 1792 | 7168 | 10752 | 8288 | 4032 | 1136 | 144 | 16 demihepteracts128 7-simplices |
Rectified 7-simplex | |||

9 | 1_{61} |
demienneract |
s{2^{1,1,1,1,1,1,1,1}}h{4,3 ^{7}}{3^{1,6,1}} |
256 | 4608 | 21504 | 37632 | 36288 | 23520 | 9888 | 2448 | 274 | 18 demiocteracts256 8-simplices |
Rectified 8-simplex | ||

10 | 1_{71} |
demidekeract |
s{2^{1,1,1,1,1,1,1,1,1}}h{4,3 ^{8}}{3^{1,7,1}} |
512 | 11520 | 61440 | 122880 | 142464 | 115584 | 64800 | 24000 | 5300 | 532 | 20 demienneracts512 9-simplices |
Rectified 9-simplex | |

... | ||||||||||||||||

n | 1_{n-3,1} |
n-demicube |
s{2^{1,1,...,1}}h{4,3 ^{n-2}}{3^{1,n-3,1}} |
2^{n-1} |
2n (n-1)-demicubes 2 ^{n-1} (n-1)-simplices |
Rectified (n-1)-simplex |

In general, a demicube's elements can be determined from the original n-cube: (With C_{n,m} = *m ^{th}*-face count in n-cube = 2

^{n-m}*n!/(m!*(n-m)!))

**Vertices:**D_{n,0}= 1/2 * C_{n,0}= 2^{n-1}(Half the n-cube vertices remain)**Edges:**D_{n,1}= C_{n,2}= 1/2 n(n-1)2^{n-2}(All original edges lost, each square faces create a new edge)**Faces:**D_{n,2}= 4 * C_{n,3}= 2/3 n(n-1)(n-2)2^{n-3}(All original faces lost, each cube creates 4 new triangular faces)**Cells:**D_{n,3}= C_{n,3}+ 2^{3}C_{n,4}(tetrahedra from original cells plus new ones)**Hypercells:**D_{n,4}= C_{n,4}+ 2^{4}C_{n,5}(16-cells and 5-cells respectively)- ...
- [For m=3...n-1]: D
_{n,m}= C_{n,m}+ 2^{m}C_{n,m+1}(m-demicubes and m-simplexes respectively) - ...
**Facets:**D_{n,n-1}= 2n + 2^{n-1}((n-1)-demicubes and (n-1)-simplices respectively)

## Symmetry group

The symmetry group of the demihypercube is the Coxeter group [3^{n-3,1,1}] has order and is an index 2 subgroup of the hyperoctahedral group (which is the Coxeter group [4,3^{n-1}]). It 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

- Regular and semi-regular polytopes III, p. 315-316
- "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: 1_{n1}) **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- (Paper 24) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3-45]

- (Paper 24) H.S.M. Coxeter,

## External links

- Olshevsky, George. "Half measure polytope".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007.