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.

Uniform 5-5 duoprism

Schlegel diagram
TypeUniform duoprism
Schläfli symbol{5}×{5} = {5}2
Coxeter diagram
Cells10 pentagonal prisms
Faces25 squares,
10 pentagons
Edges50
Vertices25
Vertex figure
Tetragonal disphenoid
Symmetry[[5,2,5]] = [10,2+,10], order 200
Dual5-5 duopyramid
Propertiesconvex, vertex-uniform, facet-transitive

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.

5-5 duoprism Penrose tiling

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.[1]


Perspective projection of complex polygon, 5{4}2 has 25 vertices and 10 5-edges, shown here with 5 red and 5 blue pentagonal 5-edges.
Orthogonal projection with coinciding central vertices
Orthogonal projection, perspective offset to avoid overlapping elements

The birectified order-5 120-cell, , constructed by all rectified 600-cells, a 5-5 duoprism vertex figure.

5-5 duopyramid
5-5 duopyramid
TypeUniform dual duopyramid
Schläfli symbol{5}+{5} = 2{5}
Coxeter diagram
Cells25 tetragonal disphenoids
Faces50 isosceles triangles
Edges35 (25+10)
Vertices10 (5+5)
Symmetry[[5,2,5]] = [10,2+,10], order 200
Dual5-5 duoprism
Propertiesconvex, vertex-uniform,
facet-transitive

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:

orthogonal projections

Two pentagons in dual positions
Two pentagons overlapping

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.[2]


Orthographic projection
The 2{4}5 with 10 vertices in blue and red connected by 25 2-edges as a complete bipartite graph.

See also

Notes
  1. Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
  2. 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.
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