# 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$_n$. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n (n−1)-demicubes, and 2$^n$ (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 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 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 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 | 1$_{k1}$ | Petrie polygon | Schläfli symbol | Coxeter diagrams A$_1$$^n$ B$_n$ D$_n$ | Elements | Elements | Elements | Elements | Elements | Elements | Elements | Elements | Elements | Elements | Facets: Demihypercubes & Simplexes | Vertex figure |
| n | 1$_{k1}$ | Petrie polygon | Schläfli symbol | Coxeter diagrams A$_1$$^n$ B$_n$ D$_n$ | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | 8-faces | 9-faces | Facets: Demihypercubes & Simplexes | Vertex figure |
| 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 demipenteracts 32 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 demihexeracts 64 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 demihepteracts 128 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 demiocteracts 256 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 demienneracts 512 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$_{n,m}$ = mth-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 stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group $BC_{n}$ [4,3$^{n−1}$]) has index 2. It is the Coxeter group $D_{n},$ [3$^{n−3,1,1}$] of order $2^{n-1}n!$, and is generated by permutations of the coordinate axes and reflections along pairs of coordinate axes.

## 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.