# 5-5 duoprism

Uniform 5-5 duoprism Schlegel diagram
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 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 [[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   ## Related complex polygons

The regular complex polytope 5{4}2,    , in $\mathbb {C} ^{2}$  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 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, 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

## 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 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 2{4}5 has 10 vertices in $\mathbb {C} ^{2}$  with a real represention in $\mathbb {R} ^{4}$  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. Orthographic projection The 2{4}5 with 10 vertices in blue and red connected by 25 2-edges as a complete bipartite graph.