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, 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 52,, in has a real representation as a 5-5 duoprism in 4-dimensional space. 52 has 25 vertices, and 10 5-edges. Its symmetry is 52, order 50. It also has a lower symmetry construction,, or 5×5, with symmetry 55, order 25. This is the symmetry if the red and blue 5-edges are considered distinct.

Perspective projection of complex polygon, 52 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

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:

Two pentagons in dual positions
Two pentagons overlapping

Related complex polygon

The regular complex polygon 25 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.

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