6-demicube

Demihexeract (6-demicube)
Petrie polygon projection
Type	Uniform 6-polytope
Family	demihypercube
Schläfli symbol	{3,33,1} = h{4,34} s{21,1,1,1,1}
Coxeter diagrams	Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
Coxeter symbol	131
5-faces	44	12 {31,2,1} 32 {34}
4-faces	252	60 {31,1,1} 192 {33}
Cells	640	160 {31,0,1} 480 {3,3}
Faces	640	{3}
Edges	240
Vertices	32
Vertex figure	Rectified 5-simplex
Symmetry group	D6, [33,1,1] = [1+,4,34] [25]+
Petrie polygon	decagon
Properties	convex

In geometry, a 6-demicube or demihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM₆ for a 6-dimensional half measure polytope.

Coxeter named this polytope as 1₃₁ from its Coxeter diagram, with a ring on one of the 1-length branches, Template:CDD. It can named similarly by a 3-dimensional exponential Schläfli symbol $\left\{3\begin{array}{c}3,3,3\\ 3\end{array}\right\}$ or {3,3^3,1}.

Cartesian coordinates

Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:

(±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

As a configuration

This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.

D6	Template:CDD	k-face	fk	f0	f1	f2	f3		f4		f5		k-figure	notes
A4	Template:CDD	( )	f0	32	15	60	20	60	15	30	6	6	r{3,3,3,3}	D6/A4 = 32*6!/5! = 32
A3A1A1	Template:CDD	{ }	f1	2	240	8	4	12	6	8	4	2	{}x{3,3}	D6/A3A1A1 = 32*6!/4!/2/2 = 240
A3A2	Template:CDD	{3}	f2	3	3	640	1	3	3	3	3	1	{3}v( )	D6/A3A2 = 32*6!/4!/3! = 640
A3A1	Template:CDD	h{4,3}	f3	4	6	4	160	*	3	0	3	0	{3}	D6/A3A1 = 32*6!/4!/2 = 160
A3A2	Template:CDD	{3,3}		4	6	4	*	480	1	2	2	1	{}v( )	D6/A3A2 = 32*6!/4!/3! = 480
D4A1	Template:CDD	h{4,3,3}	f4	8	24	32	8	8	60	*	2	0	{ }	D6/D4A1 = 32*6!/8/4!/2 = 60
A4	Template:CDD	{3,3,3}		5	10	10	0	5	*	192	1	1		D6/A4 = 32*6!/5! = 192
D5	Template:CDD	h{4,3,3,3}	f5	16	80	160	40	80	10	16	12	*	( )	D6/D5 = 32*6!/16/5! = 12
A5	Template:CDD	{3,3,3,3}		6	15	20	0	15	0	6	*	32		D6/A5 = 32*6!/6! = 32

Related polytopes

There are 47 uniform polytopes with D₆ symmetry, 31 are shared by the B₆ symmetry, and 16 are unique: Template:Demihexeract family

The 6-demicube, 1₃₁ is third in a dimensional series of uniform polytopes, expressed by Coxeter as k₃₁ series. The fifth figure is a Euclidean honeycomb, 3₃₁, and the final is a noncompact hyperbolic honeycomb, 4₃₁. Each progressive uniform polytope is constructed from the previous as its vertex figure.

k₃₁ dimensional figures

n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\tilde{E}}_7$ = E7+	${\overline{T}}_8$=E7++
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Symmetry	[3−1,3,1]	[30,3,1]	[31,3,1]	[32,3,1]	[33,3,1]	[34,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	−131	031	131	231	331	431

It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. The fourth figure is the Euclidean honeycomb 1₃₃ and the final is a noncompact hyperbolic honeycomb, 1₃₄.

1_3k dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\tilde{E}}_7$=E7+	${\overline{T}}_8$=E7++
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Symmetry	[3−1,3,1]	[30,3,1]	[31,3,1]	[32,3,1]	[[33,3,1]]	[34,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	13,-1	130	131	132	133	134

Skew icosahedron

Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.

References

• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, Template:ISBN, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Template:ISBN [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:ISBN (Chapter 26. pp. 409: Hemicubes: 1_n1)
• Template:KlitzingPolytopes

External links

• Template:GlossaryForHyperspace
• Multi-dimensional Glossary