Английская Википедия:10-simplex
| Regular hendecaxennon (10-simplex) | |
|---|---|
| Файл:10-simplex t0.svg Orthogonal projection inside Petrie polygon | |
| Type | Regular 10-polytope |
| Family | simplex |
| Schläfli symbol | {3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram | Шаблон:CDD |
| 9-faces | 11 9-simplexФайл:9-simplex t0.svg |
| 8-faces | 55 8-simplexФайл:8-simplex t0.svg |
| 7-faces | 165 7-simplexФайл:7-simplex t0.svg |
| 6-faces | 330 6-simplexФайл:6-simplex t0.svg |
| 5-faces | 462 5-simplexФайл:5-simplex t0.svg |
| 4-faces | 462 5-cellФайл:4-simplex t0.svg |
| Cells | 330 tetrahedronФайл:3-simplex t0.svg |
| Faces | 165 triangleФайл:2-simplex t0.svg |
| Edges | 55 |
| Vertices | 11 |
| Vertex figure | 9-simplex |
| Petrie polygon | hendecagon |
| Coxeter group | A10 [3,3,3,3,3,3,3,3,3] |
| Dual | Self-dual |
| Properties | convex |
In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos−1(1/10), or approximately 84.26°.
It can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn (variation of ennea for nine), having 9-dimensional facets, and -on.
Coordinates
The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are:
- <math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)</math>
- <math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)</math>
- <math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
- <math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
- <math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
- <math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
- <math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
- <math>\left(\sqrt{1/55},\ \sqrt{1/45},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
- <math>\left(\sqrt{1/55},\ -3\sqrt{1/5},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
- <math>\left(-\sqrt{20/11},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
More simply, the vertices of the 10-simplex can be positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,0,1). This construction is based on facets of the 11-orthoplex.
Images
Шаблон:A10 Coxeter plane graphs
Related polytopes
The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).
References
- Coxeter, H.S.M.:
- Шаблон:Cite book
- Шаблон:Cite book
- (Paper 22) Шаблон:Cite journal
- (Paper 23) Шаблон:Cite journal
- (Paper 24) Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite thesis
- Шаблон:KlitzingPolytopes
External links
