Uniform 1 k2 polytope
In
extended Schläfli symbol
{3,3k,2}.
Family members
The family starts uniquely as
) in 4-dimensions.Each polytope is constructed from 1k-1,2 and (n-1)-demicube facets. Each has a vertex figure of a {31,n-2,2} polytope is a birectified n-simplex, t2{3n}.
The sequence ends with k=6 (n=10), as an infinite tessellation of 9-dimensional hyperbolic space.
The complete family of 1k2 polytope polytopes are:
- 5-cell: 102, (5 tetrahedral cells)
- facets)
- demipenteractfacets)
- demihexeractfacets)
- demihepteractfacets)
- demiocteractfacets)
- demienneractfacets)
Elements
n | 1k2 | Petrie polygon projection |
Name Coxeter-Dynkin
diagram |
Facets
|
Elements | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1k-1,2 | (n-1)-demicube | Vertices | Edges | Faces | Cells
|
4-faces | 5-faces | 6-faces | 7-faces | ||||
4 | 102 | 120 |
-- | 5 110 |
5 | 10 | 10 |
5 |
|||||
5 | 112 | 121 |
16 120 |
10 111 |
16 | 80 | 160 |
120 |
26 |
||||
6 | 122 | 122 |
27 |
27 |
72 | 720 | 2160 |
2160 |
702 |
54 |
|||
7 | 132 | 132 |
56 |
126 |
576 | 10080 | 40320 |
50400 |
23688 |
4284 |
182 |
||
8 | 142 | 142 |
240 |
2160 |
17280 | 483840 | 2419200 |
3628800 |
2298240 |
725760 |
106080 |
2400 | |
9 | 152 | 152 (8-space tessellation) |
∞ |
∞ |
∞ | ||||||||
10 | 162 | 162 (9-space hyperbolic tessellation) |
∞ 152 |
∞ |
∞ |
See also
- k21 polytope family
- 2k1 polytope family
References
- Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
- Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.
- Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910.
- Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam
- Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam (eerstie sectie), vol 11.5, 1913.
- H. S. M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985
- H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988
External links
Family | An | Bn | 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
|
Pentachoron | 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: List of regular polytopes and compounds
|
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb
|
{3[4]} | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb
|
{3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb
|
{3[6]} | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb
|
{3[7]} | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb
|
{3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb
|
{3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb
|
{3[10]} | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | {3[n]}
|
δn | hδn | qδn | 1k2 • 2k1 • k21 |