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 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.

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. ... (As an alternated orthotope) s{2^n-1}
2. ... (As an alternated hypercube) h{4,3^n-1}
3. .... (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.

n	1k1	Petrie polygon	Schläfli symbol	Coxeter-Dynkin diagrams Cn family Bn 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)	s{21} h{4} {31,-1,1}		2	2									2 edges	--
3	101	demicube (tetrahedron)	s{22} h{4,3} {31,0,1}		4	6	4								(6 digons) 4 triangles	Triangle (Rectified triangle)
4	111	demitesseract (16-cell)	s{23} h{4,3,3} {31,1,1}		8	24	32	16							8 demicubes (tetrahedra) 8 tetrahedra	Octahedron (Rectified tetrahedron)
5	121	demipenteract	s{24} h{4,33}{31,2,1}		16	80	160	120	26						10 16-cells 16 5-cells	Rectified 5-cell
6	131	demihexeract	s{25} h{4,34}{31,3,1}		32	240	640	640	252	44					12 demipenteracts 32 5-simplices	Rectified hexateron
7	141	demihepteract	s{26} h{4,35}{31,4,1}		64	672	2240	2800	1624	532	78				14 demihexeracts 64 6-simplices	Rectified 6-simplex
8	151	demiocteract	s{27} h{4,36}{31,5,1}		128	1792	7168	10752	8288	4032	1136	144			16 demihepteracts 128 7-simplices	Rectified 7-simplex
9	161	demienneract	s{28} h{4,37}{31,6,1}		256	4608	21504	37632	36288	23520	9888	2448	274		18 demiocteracts 256 8-simplices	Rectified 8-simplex
10	171	demidekeract	s{29} h{4,38}{31,7,1}		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{2n-1} h{4,3n-2}{31,n-3,1}	... ... ...	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, has order 2^{n − 1}n!, and is an index 2 subgroup of the hyperoctahedral group (which is the Coxeter group BC_n).

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

• Olshevsky, George, Half measure polytope at Glossary for Hyperspace.