5-5 duoprism

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 ${\displaystyle \mathbb {C} ^{2}}$ 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
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 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

Related complex polygon

The regular complex polygon ₂{4}₅ has 10 vertices in ${\displaystyle \mathbb {C} ^{2}}$ with a real represention in ${\displaystyle \mathbb {R} ^{4}}$ 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.

• 3-3 duoprism
• 3-4 duoprism
• Tesseract (4-4 duoprism)
• Convex regular 4-polytope
• Duocylinder

• 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.

• 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