6-6 duoprism


In geometry of 4 dimensions, a 6-6 duoprism or hexagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two hexagons.
It has 36 vertices, 72 edges, 48 faces, in 12 hexagonal prism cells. It has Coxeter diagram, and symmetry, order 288.

Images

Net
Seen in a skew 2D orthogonal projection, it contains the projected rhombi of the rhombic tiling.
6-6 duoprismRhombic tiling
6-6 duoprism6-6 duoprism

Related complex polygons

The regular complex polytope 62,, in has a real representation as a 6-6 duoprism in 4-dimensional space. 62 has 36 vertices, and 12 6-edges. Its symmetry is 62, order 72. It also has a lower symmetry construction,, or 6×6, with symmetry 66, order 36. This is the symmetry if the red and blue 6-edges are considered distinct.

6-6 duopyramid

The dual of a 6-6 duoprism is called a 6-6 duopyramid or hexagonal duopyramid. It has 36 tetragonal disphenoid cells, 72 triangular faces, 48 edges, and 12 vertices.
It can be seen in orthogonal projection:
Skew

Related complex polygon

The regular complex polygon 26 or has 12 vertices in with a real representation in matching the same vertex arrangement of the 6-6 duopyramid. It has 36 2-edges corresponding to the connecting edges of the 6-6 duopyramid, while the 12 edges connecting the two hexagons 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.

Related polytopes

The 3-3 duoantiprism is an alternation of the 6-6 duoprism, but is not uniform. It has a highest symmetry construction of order 144 uniquely obtained as a direct alternation of the uniform 6-6 duoprism with an edge length ratio of 0.816 : 1. It has 30 cells composed of 12 octahedra and 18 tetrahedra. The vertex figure is a gyrobifastigium, which has a regular-faced variant that is not isogonal. It is also the convex hull of two uniform 3-3 duoprisms in opposite positions.

Vertex figure for the 3-3 duoantiprism