8-8 duoprism
Uniform 8-8 duoprism Schlegel diagram | |
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Type | Uniform duoprism |
Schläfli symbol | {8}×{8} = {8}2 |
Coxeter diagrams | |
Cells | 16 octagonal prisms |
Faces | 64 squares, 16 octagons |
Edges | 128 |
Vertices | 64 |
Vertex figure | Tetragonal disphenoid |
Symmetry | 8,2,8 = [16,2+,16], order 512 |
Dual | 8-8 duopyramid |
Properties | convex, vertex-uniform, facet-transitive |
In geometry of 4 dimensions, a 8-8 duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two octagons.
It has 64 vertices, 128 edges, 80 faces (64 squares, and 16 octagons), in 16 octagonal prism cells. It has Coxeter diagram , and symmetry 8,2,8, order 512.
Images
The uniform 8-8 duoprism can be constructed from [8]×[8] or [4]×[4] symmetry, order 256 or 64, with extended symmetry doubling these with a 2-fold rotation that maps the two orientations of prisms together. These can be expressed by 4 permutations of uniform coloring of the octahedral prism cells.
8,2,8, order 512 | [8,2,8], order 256 | 4,2,4, order 128 | [4,2,4], order 64 |
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{8}2 | {8}×{8} | t{4}2 | t{4}×t{4} |
Seen in a skew 2D orthogonal projection, it has the same vertex positions as the hexicated 7-simplex, except for a center vertex. The projected rhombi and squares are also shown in the Ammann–Beenker tiling.
8-8 duoprism |
4-4 duoprism |
Hexicated 7-simplex | |
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Ammann–Beenker tiling | {8/3} octagrams | ||
8-8 duoprism |
Related complex polygons
The regular complex polytope 8{4}2, , in has a real representation as a 8-8 duoprism in 4-dimensional space. 8{4}2 has 64 vertices, and 16 8-edges. Its symmetry is 8[4]2, order 128.
It also has a lower symmetry construction, , or 8{}×8{}, with symmetry 8[2]8, order 64. This is the symmetry if the red and blue 8-edges are considered distinct.[1]
8-8 duopyramid
8-8 duopyramid | |
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Type | Uniform dual duopyramid |
Schläfli symbol | {8}+{8} = 2{8} |
Coxeter diagrams | |
Cells | 64 tetragonal disphenoids |
Faces | 128 isosceles triangles |
Edges | 80 (64+16) |
Vertices | 16 (8+8) |
Symmetry | 8,2,8 = [16,2+,16], order 512 |
Dual | 8-8 duoprism |
Properties | convex, vertex-uniform, facet-transitive |
The dual of a 8-8 duoprism is called a 8-8 duopyramid. It has 64 tetragonal disphenoid cells, 128 triangular faces, 80 edges, and 16 vertices.
Skew | [16] |
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Related complex polygon
The regular complex polygon 2{4}8 has 16 vertices in with a real represention in matching the same vertex arrangement of the 8-8 duopyramid. It has 64 2-edges corresponding to the connecting edges of the 8-8 duopyramid, while the 16 edges connecting the two octagons are not included.
The vertices and edges makes a complete bipartite graph with each vertex from one octagon is connected to every vertex on the other.[2]
See also
- 3-3 duoprism
- 3-4 duoprism
- 5-5 duoprism
- Tesseract (4-4 duoprism)
- Convex regular 4-polytope
- Duocylinder
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
- Olshevsky, George. "Duoprism". Glossary for Hyperspace. Archived from the original on 4 February 2007.
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