In geometry, demihypercubes (also called ndemicubes, nhemicubes, and half measure polytopes) are a class of npolytopes constructed from alternation of an nhypercube, 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.^{[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 16cell. 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 ndemicube 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 ndimensions above three. He called it a 5ic semiregular. 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 nsimplexes.
They are represented by CoxeterDynkin diagrams of three constructive forms:
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 ndemicube, 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  4faces  5faces  6faces  7faces  8faces  9faces  
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 (16cell) 
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 16cells 16 5cells 
Rectified 5cell  
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 5simplices 
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 6simplices 
Rectified 6simplex  
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 7simplices 
Rectified 7simplex  
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 8simplices 
Rectified 8simplex  
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 9simplices 
Rectified 9simplex 
...  
n  1_{n−3,1}  ndemicube  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 ncube: (with C_{n,m} = m^{th}face count in ncube = 2^{n−m} n!/(m!(n−m)!))
The stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group [4,3^{n−1}]) has index 2. It is the Coxeter group [3^{n−3,1,1}] of order , and is generated by permutations of the coordinate axes and reflections along pairs of coordinate axes.^{[2]}
Constructions as alternated orthotopes have the same topology, but can be stretched with different lengths in naxes of symmetry.
The rhombic disphenoid is the threedimensional example as alternated cuboid. It has three sets of edge lengths, and scalene triangle faces.