Английская Википедия:Great duoantiprism
| Great duoantiprism | |
|---|---|
| Type | Uniform polychoron |
| Schläfli symbols | Шаблон:Math |
| Coxeter diagrams | Шаблон:CDD Шаблон:CDD Шаблон:CDD Шаблон:CDD |
| Cells | 50 tetrahedra Файл:Tetrahedron.png 10 pentagonal antiprisms Файл:Pentagonal antiprism.png 10 pentagrammic crossed-antiprisms Файл:Pentagrammic crossed antiprism.png |
| Faces | 200 triangles 10 pentagons 10 pentagrams |
| Edges | 200 |
| Vertices | 50 |
| Vertex figure | Файл:Great duoantiprism verf.png star-gyrobifastigium |
| Symmetry group | Шаблон:Math order 50 Шаблон:Math order 100 Шаблон:Math order 200 |
| Properties | Vertex-uniform |
| Файл:Great duoantiprism net.png Net (overlapping in space) | |
In geometry, the great duoantiprism is the only uniform star-duoantiprism solution Шаблон:Math Шаблон:Math in 4-dimensional geometry. It has Schläfli symbol Шаблон:Math Шаблон:Math or Шаблон:Math Coxeter diagram Шаблон:CDD, constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra.
Its vertices are a subset of those of the small stellated 120-cell.
Construction
The great duoantiprism can be constructed from a nonuniform variant of the 10-10/3 duoprism (a duoprism of a decagon and a decagram) where the decagram's edge length is around 1.618 (golden ratio) times the edge length of the decagon via an alternation process. The decagonal prisms alternate into pentagonal antiprisms, the decagrammic prisms alternate into pentagrammic crossed-antiprisms with new regular tetrahedra created at the deleted vertices. This is the only uniform solution for the p-q duoantiprism aside from the regular 16-cell (as a 2-2 duoantiprism).
Images
| Файл:Great duoantiprism.png stereographic projection, centered on one pentagrammic crossed-antiprism |
Файл:Gudap orthogonal projection.png Orthogonal projection, with vertices colored by overlaps, red, orange, yellow, green have 1, 2, 3,4 multiplicity. |
Other names
References
- Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Шаблон:KlitzingPolytopes
- ↑ Jonathan Bowers - Miscellaneous Uniform Polychora 965. Gudap
- ↑ http://www.polychora.com/12GudapsMovie.gif Animation of cross sections
