6-6 duoprism
Uniform 6-6 duoprism Schlegel diagram | |
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Type | Uniform duoprism |
Schläfli symbol | {6}×{6} = {6}2 |
Coxeter diagrams | |
Cells | 12 hexagonal prisms |
Faces | 36 squares, 12 hexagons |
Edges | 72 |
Vertices | 36 |
Vertex figure | Tetragonal disphenoid |
Symmetry | [[6,2,6]] = [12,2+,12], order 288 |
Dual | 6-6 duopyramid |
Properties | convex, vertex-uniform, facet-transitive |
In geometry of 4 dimensions, a 6-6 duoprism or hexagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two hexagons.
It has 36 vertices, 72 edges, 48 faces (36 squares, and 12 hexagons), in 12 hexagonal prism cells. It has Coxeter diagram
Images
Net
Seen in a skew 2D orthogonal projection, it contains the projected rhombi of the rhombic tiling.
6-6 duoprism | Rhombic tiling |
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6-6 duoprism | 6-6 duoprism |
Related complex polygons
The regular complex polytope 6{4}2,
6-6 duopyramid
6-6 duopyramid | |
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Type | Uniform dual duopyramid |
Schläfli symbol | {6}+{6} = 2{6} |
Coxeter diagrams | |
Cells | 36 tetragonal disphenoids |
Faces | 72 isosceles triangles |
Edges | 48 (36+12) |
Vertices | 12 (6+6) |
Symmetry | [[6,2,6]] = [12,2+,12], order 288 |
Dual | 6-6 duoprism |
Properties | convex, vertex-uniform, facet-transitive |
The dual of a 6-6 duoprism is called a 6-6 duopyramid or hexagonal duopyramid. It has 36 tetragonal disphenoid cells, 72 triangular faces, 48 edges, and 12 vertices.
It can be seen in orthogonal projection:
Skew | [6] | [12] |
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Related complex polygon
The regular complex polygon 2{4}6 or
The vertices and edges makes a complete bipartite graph with each vertex from one pentagon is connected to every vertex on the other.[2]
See also
Notes
- ↑ Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
- ↑ Regular Complex Polytopes, p.114
References
- Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
- Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999,
ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
- Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Catalogue of Convex Polychora, section 6, George Olshevsky.
External links
- The Fourth Dimension Simply Explained—describes duoprisms as "double prisms" and duocylinders as "double cylinders"
- Polygloss - glossary of higher-dimensional terms
- Exploring Hyperspace with the Geometric Product