Demihypercube

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{4,3,...,3} as half the vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes.

They are represented by Coxeter-Dynkin diagrams of three constructive forms:

1. {{CDD|node_h|2c|node_h|2c|node_h}}...{{CDD|node_h}} (As an alternated orthotope) s{2^1,1,...,1}
2. {{CDD|node_h|4|node|3}}...{{CDD|3|node}} (As an alternated hypercube) h{4,3ⁿ⁻¹}
3. {{CDD|nodes_10ru|split2|node|3}}...{{CDD|3|node}}. (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.

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 = 2ⁿ⁻¹ (Half the n-cube vertices remain)
• Edges: D_n,1 = C_n,2 = 1/2 n(n−1) 2ⁿ⁻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) 2ⁿ⁻³ (All original faces lost, each cube creates 4 new triangular faces)
• Cells: D_n,3 = C_n,3 + 2³ C_n,4 (tetrahedra from original cells plus new ones)
• Hypercells: D_n,4 = C_n,4 + 2⁴ 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 + 2ⁿ⁻¹ ((n−1)-demicubes and (n−1)-simplices respectively)

The stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group $BC\_n$ [4,3ⁿ⁻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.{{cite web|url=http://math.ucr.edu/home/baez/week187.html|title=week187|website=math.ucr.edu|access-date=20 April 2018}}

File:Rhombic disphenoid.png]]

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.

• Hypercube honeycomb
• Semiregular E-polytope

{{Reflist}}

• 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-Strauss, 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

• {{GlossaryForHyperspace | anchor=half | title=Half measure polytope }}

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{{Polytopes}}

Category:Multi-dimensional geometry

Category:Uniform polytopes