User:Tomruen/3-3-3 prism
 Orthogonal projections in regular enneagon | |
| Type | p-q-r prism |
| Schläfli symbol | {3}×{3}×{3} = {3}3 |
| Coxeter diagram |     or 
|
| 5-faces | 9 {3}×{3}×{ } |
| 4-faces | 9 {3}×{3} 27 {3}×{4} |
| Cells | 54 {3}×{ } 27 {4}×{ } |
| Faces | 81 {4} 27 {3} |
| Edges | 81 |
| Vertices | 27 |
| Vertex figure | 5-simplex
|
| Symmetry | [3[3,2,3,2,3]], order 64 =1296 |
| Dual | 3-3-3 pyramid |
| Properties | convex, vertex-uniform, facet-transitive |
In the geometry of 6 dimensions, the 3-3-3 prism or triangular triaprism is a four-dimensional convex uniform polytope. It can be constructed as the Cartesian product of three triangles and is the simplest of an infinite family of six-dimensional polytopes constructed as Cartesian products of three polygons.
Elements
[edit]It has 27 vertices, 81 edges, 108 faces (81 squares, and 27 triangles), 54 triangular prism,{3}×{ }, 27 square prisms, { }×{ }×{ }, and 9 3-3 duoprisms, {3}×{3} ,27 3-4 duoprisms, {3}×{4}, and 18, 3-3 duoprism prisms, {3}×{3}×{ }.[1] It has Coxeter diagram , and Coxeter notation symmetry [3[3,2,3,2,3]], order 1296. The symmetry of each triangle is [3], dihedral order 6. All three triangles combined have symmetry order 63 = 216. The extended symmetry [3], order 6 comes from permuting the three planes of triangles.
Its vertex figure is a 6-simplex with 3 orthogonal longer edges.
Projections
[edit] Orthogonal projections in regular enneagon Upper left, edge colored by triangle. Others show 3 sets of 9 triangles. |
Related figures
[edit]The 3-3-3 prism shares vertices with a generalized cube, a complex polyhedron, 3{4}2{3}2, 
or 
, with 27 vertices, 27 3-edges, and 9 faces.
The 3-3-3 prism is the vertex figure of the birectified 222 honeycomb, 0222, {32,2,2}, or 
in 6-dimensions.
The 4-4-4 prism is the same as a 6-cube.
References
[edit]- ^ Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o x3o x3o - trittip
- 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