Small stellated 120-cell
| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Small stellated 120-cell | |
| bgcolor=#ffffff align=center colspan=2|280px Orthogonal projection | |
| bgcolor=#e7dcc3|Type | Schläfli-Hess polytope |
| bgcolor=#e7dcc3|Cells | 120 {5/2,5} |
| bgcolor=#e7dcc3|Faces | 720 {5/2} |
| bgcolor=#e7dcc3|Edges | 1200 |
| bgcolor=#e7dcc3|Vertices | 120 |
| bgcolor=#e7dcc3|Vertex figure | {5,3} |
| bgcolor=#e7dcc3|Schläfli symbol | {5/2,5,3} |
| bgcolor=#e7dcc3|Coxeter-Dynkin diagram | {{CDD|node_1|5|rat|d2|node|5|node|3|node}} |
| bgcolor=#e7dcc3|Symmetry group | H4, [3,3,5] |
| bgcolor=#e7dcc3|Dual | Icosahedral 120-cell |
| bgcolor=#e7dcc3|Properties | Regular |
In geometry, the small stellated 120-cell or stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,5,3}. It is one of 10 regular Schläfli-Hess polytopes.
Related polytopes
It has the same edge arrangement as the great grand 120-cell, and also shares its 120 vertices with the 600-cell and eight other regular star 4-polytopes. It may also be seen as the first stellation of the 120-cell. In this sense it could be seen as analogous to the three-dimensional small stellated dodecahedron, which is the first stellation of the dodecahedron.{{cite journal
| last1 = Conrad | first1 = J.
| last2 = Chamberland | first2 = C.
| last3 = Breuckmann | first3 = N. P.
| last4 = Terhal | first4 = B. M.
| journal = Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
| volume = 376 | issue = 2123 | language = en
| issn = 1364-503X
| year = 2018
| pages = 20170323
| doi = 10.1098/rsta.2017.0323
| title = The small stellated dodecahedron code and friends
| pmid = 29807900
| pmc = 5990658
| bibcode = 2018RSPTA.37670323C
}} Indeed, the small stellated 120-cell is dual to the icosahedral 120-cell, which could be taken as a 4D analogue of the great dodecahedron, dual of the small stellated dodecahedron.
The edges of the small stellated 120-cell are τ2 as long as those of the 120-cell core inside the 4-polytope.
| class="wikitable" width=600
|+ Orthographic projections by Coxeter planes |
| align=center
!H3 !A2 / B3 / D4 !A3 / B2 |
| align=center |
See also
- List of regular polytopes
- Convex regular 4-polytope - Set of convex regular 4-polytope
- Kepler-Poinsot solids - regular star polyhedron
- Star polygon - regular star polygons
References
{{Reflist}}
- Edmund Hess, (1883) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001].
- H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}.
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} (Chapter 26, Regular Star-polytopes, pp. 404–408)
- {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|o3o5o5/2x - sishi}}
External links
- [http://hometown.aol.com/hedrondude/regulars.html Regular polychora]
- [http://mathforum.org/library/drmath/view/54786.html Discussion on names]
- [https://web.archive.org/web/20061107052613/http://www.mathematik.uni-regensburg.de/Goette/sterne/ Reguläre Polytope]
- [https://web.archive.org/web/20070704012333/http://davidf.faricy.net/polyhedra/Star_Polychora.html The Regular Star Polychora]
- [http://homepages.wmich.edu/~drichter/finalstellation120cell.htm Zome Model of the Final Stellation of the 120-cell]
- [http://homepages.wmich.edu/~drichter/stellated120cell01.htm The First Stellation of the 120-cell, A Zome Model]
{{Regular 4-polytopes}}
{{polychora-stub}}