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
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.
h{4,3,3,3} |
h2{4,3,3,3} |
h3{4,3,3,3} |
h4{4,3,3,3} |
h2,3{4,3,3,3} |
h2,4{4,3,3,3} |
h3,4{4,3,3,3} |
h2,3,4{4,3,3,3} |
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.
En | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|
Coxeter group |
E3=A2×A1 | E4=A4 | E5=D5 | E6 | E7 | E8 | E9 = = 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 |
n | 4 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|
Coxeter group |
E3=A2×A1 | E4=A4 | E5=D5 | E6 | E7 | E8 | E9 = = 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
- (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: 1n1)
- Richard Klitzing, 5D uniform polytopes (polytera), x3o3o *b3o3o - hin
External links
- Olshevsky, George, Demipenteract at Glossary for Hyperspace.
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
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Family | An | BCn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform polychoron | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes |