5-demicube

Demipenteract (5-demicube)
File:Demipenteract graph ortho.svg Petrie polygon projection
Type	Uniform 5-polytope
Family (Dn)	5-demicube
Families (En)	k21 polytope 1k2 polytope
Coxeter symbol	121
Schläfli symbols	{3,32,1} = h{4,33} s{2,4,3,3} or h{2}h{4,3,3} sr{2,2,4,3} or h{2}h{2}h{4,3} h{2}h{2}h{2}h{4} s{21,1,1,1} or h{2}h{2}h{2}s{2}
Coxeter diagrams	File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.png
4-faces	26	10 {31,1,1} File:Cross graph 4.svg 16 {3,3,3} File:4-simplex t0.svg
Cells	120	40 {31,0,1} File:3-simplex t0.svg 80 {3,3} File:3-simplex t0.svg
Faces	160	{3} File:2-simplex t0.svg
Edges	80
Vertices	16
Vertex figure	Rectified 5-cell File:5-demicube verf.svg
Petrie polygon	Octagon
Symmetry	D5, [32,1,1] = [1+,4,33] [24]+
Properties	convex

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular facets), he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM₅ for a 5-dimensional half measure polytope.

Coxeter named this polytope as 1₂₁ from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.png and Schläfli symbol $\left\{3\begin{array}{c}3,3\\ 3\end{array}\right\}$ or {3,3^2,1}.

It exists in the k₂₁ polytope family as 1₂₁ with the Gosset polytopes: 2₂₁, 3₂₁, and 4₂₁.

The graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead.

Cartesian coordinates

Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:

(±1,±1,±1,±1,±1)

with an odd number of plus signs.

As a configuration

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

D5	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	k-face	fk	f0	f1	f2	f3		f4		k-figure	Notes(*)
A4	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	( )	f0	16	10	30	10	20	5	5	rectified 5-cell	D5/A4 = 16*5!/5! = 16
A2A1A1	File:CDel nodea 1.pngFile:CDel 2.pngFile:CDel nodes x0.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	{ }	f1	2	80	6	3	6	3	2	triangular prism	D5/A2A1A1 = 16*5!/3!/2/2 = 80
A2A1	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodes 0x.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	{3}	f2	3	3	160	1	2	2	1	Isosceles triangle	D5/A2A1 = 16*5!/3!/2 = 160
A3A1	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	h{4,3}	f3	4	6	4	40	*	2	0	Segment { }	D5/A3A1 = 16*5!/4!/2 = 40
A3	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodes 0x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	{3,3}		4	6	4	*	80	1	1	Segment { }	D5/A3 = 16*5!/4! = 80
D4	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	h{4,3,3}	f4	8	24	32	8	8	10	*	Point ( )	D5/D4 = 16*5!/8/4! = 10
A4	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodes 0x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	{3,3,3}		5	10	10	0	5	*	16	Point ( )	D5/A4 = 16*5!/5! = 16

* = The number of elements (diagonal values) can be computed by the symmetry order D₅ divided by the symmetry order of the subgroup with selected mirrors removed.

Projected images

File:Demipenteract wf.png Perspective projection.

Images

orthographic projections

Coxeter plane	B5
Graph	File:5-demicube t0 B5.svg
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph	File:5-demicube t0 D5.svg	File:5-demicube t0 D4.svg
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph	File:5-demicube t0 D3.svg	File:5-demicube t0 A3.svg
Dihedral symmetry	[4]	[4]

Related polytopes

It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 Uniform 5-polytopes (uniform 5-polytopes) that can be constructed from the D₅ symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.

The 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (5-simplices and 5-orthoplexes in the case of the 5-demicube). In Coxeter's notation the 5-demicube is given the symbol 1₂₁.

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover, New York, , 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, wiley.com,
    • (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, (Chapter 26. p. 409: Hemicubes: 1_n1)

External links

• Multi-dimensional Glossary