5-5 duoprism
Uniform 5-5 duoprism | |
|---|---|
| Type | Uniform duoprism |
| Schläfli symbol | {5}×{5} = {5}2 |
| Coxeter diagram |     |
| Cells | 10 pentagonal prisms |
| Faces | 25 squares, 10 pentagons |
| Edges | 50 |
| Vertices | 25 |
| Vertex figure |  Tetragonal disphenoid |
| Symmetry | [[5,2,5]] = [10,2+,10], order 200 |
| Dual | 5-5 duopyramid |
| Properties | convex, vertex-uniform, facet-transitive |
In geometry of 4 dimensions, a 5-5 duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons.
It has 25 vertices, 50 edges, 35 faces (25 squares, and 10 pentagons), in 10 pentagonal prism cells. It has Coxeter diagram , and symmetry [[5,2,5]], order 200.
Images
 Orthogonal projection |
 Orthogonal projection |
 Net |
Seen in a skew 2D orthogonal projection, 20 of the vertices are in two decagonal rings, while 5 project into the center. The 5-5 duoprism here has an identical 2D projective appearance to the 3D rhombic triacontahedron. In this projection, the square faces project into wide and narrow rhombi seen in penrose tiling.
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| 5-5 duoprism | Penrose tiling | |
|---|---|---|
Related honeycombs and polytopes
The birectified order-5 120-cell, 
, constructed by all rectified 600-cells, a 5-5 duoprism vertex figure.
5-5 duopyramid
| 5-5 duopyramid | |
|---|---|
| Type | Uniform dual duopyramid |
| Schläfli symbol | {5}+{5} = 2{5} |
| Coxeter diagram |  |
| Cells | 25 tetragonal disphenoids |
| Faces | 50 isosceles triangles |
| Edges | 35 (25+10) |
| Vertices | 10 (5+5) |
| Symmetry | [[5,2,5]] = [10,2+,10], order 200 |
| Dual | 5-5 duoprism |
| Properties | convex, vertex-uniform, facet-transitive |
The dual of a 5-5 duoprism is called a 5-5 duopyramid. It has 25 tetragonal disphenoid cells, 50 triangular faces, 35 edges, and 10 vertices.
It can be seen in orthogonal projection as a regular 10-gon circle of vertices, divided into two pentagons, seen with colored vertices and edges:
 Two pentagons in dual positions |
 Two pentagons overlapping |
See also
- 3-3 duoprism
- 3-4 duoprism
- Tesseract (4-4 duoprism)
- Convex regular 4-polytope
- Duocylinder
Notes
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 at Glossary for Hyperspace.
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