5-5 duoprism

In geometry of 4 dimensions, a 5-5 duoprism or pentagonal 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 , order 200.

Images

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.

Related complex polygons

The regular complex polytope 5{4}2, , in  has a real representation as a 5-5 duoprism in 4-dimensional space. 5{4}2 has 25 vertices, and 10 5-edges. Its symmetry is 5[4]2, order 50. It also has a lower symmetry construction, , or 5{}×5{}, with symmetry 5[2]5, order 25. This is the symmetry if the red and blue 5-edges are considered distinct.

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

The dual of a 5-5 duoprism is called a 5-5 duopyramid or pentagonal 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:

Related complex polygon

The regular complex polygon 2{4}5 has 10 vertices in  with a real represention in  matching the same vertex arrangement of the 5-5 duopyramid. It has 25 2-edges corresponding to the connecting edges of the 5-5 duopyramid, while the 10 edges connecting the two pentagons are not included. The vertices and edges makes a complete bipartite graph with each vertex from one pentagon is connected to every vertex on the other.

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,    (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,  (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

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

Category:Polychora