Demihypercube

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In geometry, demihypercubes (also called n-demicubes, 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 are don't have all regular facets but are all uniform polytopes.

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 E-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 two interchangeable constructive forms: ... = .... H.S.M. Coxeter also labeled these bifurcating figures as 1_k,1 representing the lengths of the 3 branches and lead by the ringed branch.

n	hγn	1k1	Graph	Name 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
2	hγ2	1-1,1		digon h{4}={2}		2	2								2 edges	--
3	hγ3	101		demicube (Same as tetrahedron) h{4,3}={3,3}		4	6	4							(6 digons) 4 triangles	Triangle (Rectified triangle)
4	hγ4	111		demitesseract (Same as 16-cell) h{4,3,3}={3,3,4}		8	24	32	16						8 demicubes (tetrahedra) 8 tetrahedra	Octahedron (Rectified tetrahedron)
5	hγ5	121		demipenteract h{4,33}		16	80	160	120	26					10 16-cells 16 5-cells	Rectified 5-cell
6	hγ6	131		demihexeract h{4,34}		32	240	640	640	252	44				12 demipenteracts 32 5-simplices	Rectified hexateron
7	hγ7	141		demihepteract h{4,35}		64	672	2240	2800	1624	532	78			14 demihexeracts 64 6-simplices	Rectified heptapeton
8	hγ8	151		demiocteract h{4,36}		128	1792	7168	10752	8288	4032	1136	144		16 demihepteracts 128 7-simplices	Rectified octaexon
9	hγ9	161		demienneract h{4,37}		256	4608	21504	37632	36288	23520	9888	2448	274	18 demiocteracts 256 8-simplices	Rectified enneazetton
...
n	hγn	1n-3,1		n-demicube h{4,3n-2}	... ...	2n-1	n(n-1)2n-3	?	?	?	?	?	?	?	2n (n-1)-demicubes 2n (n-1)-simplices	Rectified (n-1)-simplex

[edit] See also

• Hypercube honeycomb

[edit] References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900

[edit] External links

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