Demihypercube should not be confused with Hemicube (geometry).
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 (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 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.
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 as half the vertices of . The vertex figures of demihypercubes are rectified n-simplexes.
They are represented by Coxeter-Dynkin diagrams of three constructive forms:
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
| 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 h | width=150 | 2 | 2 | 2 edges | -- | |||||||||
| 3 | 101 | demicube (tetrahedron) | s h | 4 | 6 | 4 | (6 digons) 4 triangles | Triangle (Rectified triangle) | |||||||||
| 111 | demitesseract (16-cell) | s h | 8 | 24 | 32 | 16 | 8 demicubes (tetrahedra) 8 tetrahedra | Octahedron (Rectified tetrahedron) | |||||||||
| 121 | demipenteract | s h | 16 | 80 | 160 | 120 | 26 | 10 16-cells 16 5-cells | Rectified 5-cell | ||||||||
| 6 | 131 | demihexeract | s h | 32 | 240 | 640 | 640 | 252 | 44 | 12 demipenteracts 32 5-simplices | Rectified hexateron | ||||||
| 7 | 141 | demihepteract | s h | 64 | 672 | 2240 | 2800 | 1624 | 532 | 78 | 14 demihexeracts 64 6-simplices | Rectified 6-simplex | |||||
| 8 | 151 | demiocteract | s h | 128 | 1792 | 7168 | 10752 | 8288 | 4032 | 1136 | 144 | 16 demihepteracts 128 7-simplices | Rectified 7-simplex | ||||
| 9 | 161 | demienneract | s h | 256 | 4608 | 21504 | 37632 | 36288 | 23520 | 9888 | 2448 | 274 | 18 demiocteracts 256 8-simplices | Rectified 8-simplex | |||
| 10 | 171 | demidekeract | s h | 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 h | ... ... ... | 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)!))
BCn
Dn,
2n-1n!
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.