5-demicube

Demipenteract (5-demicube)
Petrie polygon projection
Type	Uniform 5-polytope
Family (Dn)	5-demicube
Families (En)	k21 polytope 1k2 polytope
Coxeter symbol	121
Schläfli symbol	{3,32,1} = h{4,33} s{24}
Coxeter-Dynkin diagram
4-faces	26	10 {31,1,1} 16 {3,3,3}
Cells	120	40 {31,0,1} 80 {3,3}
Faces	160	{3}
Edges	80
Vertices	16
Vertex figure	rectified 5-cell
Petrie polygon	Octagon
Symmetry group	D5, [34,1,1] = [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 truncated.

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.

Coxeter named this polytope as 1₂₁ from its Coxeter-Dynkin diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches. It exists in the k₂₁ polytope family as 1₂₁ with the Gosset polytopes: 2₂₁, 3₂₁, and 4₂₁.

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.

Projected images

Perspective projection.

Images

orthographic projections

Coxeter plane	B5
Graph
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph
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 polytera (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.

t0(121)

t0,1(121)

t0,2(121)

t0,3(121)

t0,1,2(121)

t0,1,3(121)

t0,2,3(121)

t0,1,2,3(121)

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-cells and 16-cells in the case of the rectified 5-cell). In Coxeter's notation the 5-demicube is given the symbol 1₂₁.

k₂₁ figures in n dimensional

En	3	4	5	6	7	8	9	10
Coxeter group	E3=A2×A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde{E}}_{8}$ = E8+	E10 = E8++
Coxeter diagram
Symmetry (order)	[3-1,2,1] (12)	[30,2,1] (120)	[31,2,1] (192)	[32,2,1] (51,840)	[33,2,1] (2,903,040)	[34,2,1] (696,729,600)	[35,2,1] (∞)	[36,2,1] (∞)
Graph							∞	∞
Name	−121	021	121	221	321	421	521	621

1_k2 figures in n dimensions

n	4	4	5	6	7	8	9	10
Coxeter group	E3=A2×A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde{E}}_{8}$ = E8+	E10 = E8++
Coxeter diagram
Symmetry (order)	[3-1,2,1] (12)	[30,2,1] (120)	[31,2,1] (192)	[[32,2,1]] (103,680)	[33,2,1] (2,903,040)	[34,2,1] (696,729,600)	[35,2,1] (∞)	[36,2,1] (∞)
Graph							∞	∞
Name	1-1,2	102	112	122	132	142	152	162

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 edition, ISBN 0-486-61480-8, 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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
• Richard Klitzing, 5D uniform polytopes (polytera), x3o3o *b3o3o - hin

External links

• Olshevsky, George, Demipenteract at Glossary for Hyperspace.
• Multi-dimensional Glossary