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 (36 squares, and 12 hexagons), in 12 hexagonal prism cells. It has Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png, and symmetry [[6,2,6]], order 288.

Seen in a skew 2D orthogonal projection, it contains the projected rhombi of the rhombic tiling.

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.

The vertices and edges makes a complete bipartite graph with each vertex from one pentagon is connected to every vertex on the other.[2]

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 (as triangular antiprisms) and 18 tetrahedra (as tetragonal disphenoids). 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.