Uniform k 21 polytope
In
Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the 5-ic semiregular figure.
Family members
The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the
The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness. The demipenteract also exists in the demihypercube family.
They are also sometimes named by their symmetry group, like E6 polytope, although there are many uniform polytopes within the E6 symmetry.
The complete family of Gosset semiregular polytopes are:
- squarefaces)
- rectified 5-cell: 021, Tetroctahedric (5 tetrahedra and 5 octahedra cells)
- facets)
- orthoplexfacets)
- orthoplexfacets)
- orthoplexfacets)
- orthoplexfacets)
- orthoplexfacets)
Each polytope is constructed from (n − 1)-
The orthoplex faces are constructed from the
Each has a vertex figure as the previous form. For example, the rectified 5-cell has a vertex figure as a triangular prism.
Elements
n-ic | k21 | Graph | Name Coxeter
diagram |
Facets
|
Elements | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(n − 1)-simplex {3n−2} |
(n − 1)- orthoplex {3n−4,1,1} |
Vertices | Edges | Faces | Cells
|
4-faces | 5-faces | 6-faces | 7-faces | ||||
3-ic | −121 | Triangular prism |
2 triangles |
3 |
6 | 9 | 5 | ||||||
4-ic | 021 | Rectified 5-cell |
5 tetrahedron |
5 octahedron |
10 | 30 | 30 | 10 | |||||
5-ic | 121 | Demipenteract |
16 5-cell |
10 16-cell |
16 | 80 | 160 | 120 | 26 | ||||
6-ic | 221 | 221 polytope |
72 5-simplexes |
27 5-orthoplexes |
27 | 216 | 720 | 1080 | 648 | 99 | |||
7-ic | 321 | 321 polytope |
576 6-simplexes |
126 6-orthoplexes |
56 | 756 | 4032 | 10080 | 12096 | 6048 | 702 | ||
8-ic | 421 | 421 polytope |
17280 7-simplexes |
2160 7-orthoplexes |
240 | 6720 | 60480 | 241920 | 483840 | 483840 | 207360 | 19440 | |
9-ic | 521 | 521 honeycomb |
∞ 8-simplexes |
∞ 8-orthoplexes |
∞ | ||||||||
10-ic | 621 | 621 honeycomb |
∞ 9-simplexes |
∞ 9-orthoplexes |
∞ |
See also
- Uniform 2k1 polytope family
- Uniform 1k2 polytope family
References
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
- 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
- G.Blind and R.Blind, "The semi-regular polyhedra", Commentari Mathematici Helvetici 66 (1991) 150–154
- ISBN 978-1-56881-220-5(Chapter 26. pp. 411–413: The Gosset Series: n21)
External links
- PolyGloss v0.05: Gosset figures (Gossetoicosatope)
- Regular, SemiRegular, Regular faced and Archimedean polytopes Archived 2011-07-19 at the Wayback Machine
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 |