Root datum

Alternation of the n-cube yields one of two n-demicubes, as in this 3-dimensional illustration of the two tetrahedra that arise as the 3-demicubes of the 3-cube.

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 2ⁿ (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 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.

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 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.

Constructions

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

1. Template:CDD...Template:CDD (As an alternated orthotope) sr{2^n-1}
2. Template:CDD...Template:CDD (As an alternated hypercube) h{4,3^n-1}
3. Template:CDD...Template:CDD. (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 lead 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-Dynkin diagrams A1n family BCn family Dn family	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)	sr{21} h{4} {31,-1,1}	Template:CDD Template:CDD Template:CDD	2	2									2 edges	--
3	101	demicube (tetrahedron)	sr{22} h{4,3} {31,0,1}	Template:CDD Template:CDD Template:CDD	4	6	4								(6 digons) 4 triangles	Triangle (Rectified triangle)
4	111	demitesseract (16-cell)	sr{23} h{4,3,3} {31,1,1}	Template:CDD Template:CDD Template:CDD	8	24	32	16							8 demicubes (tetrahedra) 8 tetrahedra	Octahedron (Rectified tetrahedron)
5	121	demipenteract	sr{24} h{4,33}{31,2,1}	Template:CDD Template:CDD Template:CDD	16	80	160	120	26						10 16-cells 16 5-cells	Rectified 5-cell
6	131	demihexeract	sr{25} h{4,34}{31,3,1}	Template:CDD Template:CDD Template:CDD	32	240	640	640	252	44					12 demipenteracts 32 5-simplices	Rectified hexateron
7	141	demihepteract	sr{26} h{4,35}{31,4,1}	Template:CDD Template:CDD Template:CDD	64	672	2240	2800	1624	532	78				14 demihexeracts 64 6-simplices	Rectified 6-simplex
8	151	demiocteract	sr{27} h{4,36}{31,5,1}	Template:CDD Template:CDD Template:CDD	128	1792	7168	10752	8288	4032	1136	144			16 demihepteracts 128 7-simplices	Rectified 7-simplex
9	161	demienneract	sr{28} h{4,37}{31,6,1}	Template:CDD Template:CDD Template:CDD	256	4608	21504	37632	36288	23520	9888	2448	274		18 demiocteracts 256 8-simplices	Rectified 8-simplex
10	171	demidekeract	sr{29} h{4,38}{31,7,1}	Template:CDD Template:CDD Template:CDD	512	11520	61440	122880	142464	115584	64800	24000	5300	532	20 demienneracts 512 9-simplices	Rectified 9-simplex
...
n	1n-3,1	n-demicube	sr{2n-1} h{4,3n-2}{31,n-3,1}	Template:CDD...Template:CDD Template:CDD...Template:CDD Template:CDD...Template:CDD	2n-1										n (n-1)-demicubes 2n (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 = 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^n-4C_n,4 (tetrahedra from original cells plus new ones)
• Hypercells: D_n,4 = C_n,4 + 2^n-5C_n,5 (16-cells and 5-cells respectively)
• ...
• [For m=3...n-1]: D_n,m = C_n,m + 2^n-1-mC_n,m+1 (m-demicubes and m-simplexes respectively)
• ...
• Facets: D_n,n-1 = n + 2ⁿ ((n-1)-demicubes and (n-1)-simplices respectively)

Symmetry group

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

Orthotopic constructions

The rhombic disphenoid inside of a cuboid

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.

See also

• Hypercube honeycomb
• Semiregular E-polytope

References

• 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)

External links

• Template:GlossaryForHyperspace