8-8 duoprism


In geometry of 4 dimensions, a 8-8 duoprism or octagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two octagons.
It has 64 vertices, 128 edges, 80 faces, in 16 octagonal prism cells. It has Coxeter diagram, and symmetry, order 512.

Images

The uniform 8-8 duoprism can be constructed from × or × symmetry, order 256 or 64, with extended symmetry doubling these with a 2-fold rotation that maps the two orientations of prisms together. These can be expressed by 4 permutations of uniform coloring of the octagonal prism cells.
Seen in a skew 2D orthogonal projection, it has the same vertex positions as the hexicated 7-simplex, except for a center vertex. The projected rhombi and squares are also shown in the Ammann–Beenker tiling.

Related complex polygons

The regular complex polytope 82,, in has a real representation as an 8-8 duoprism in 4-dimensional space. 82 has 64 vertices, and 16 8-edges. Its symmetry is 82, order 128.
It also has a lower symmetry construction,, or 8×8, with symmetry 88, order 64. This is the symmetry if the red and blue 8-edges are considered distinct.

8-8 duopyramid

The dual of a 8-8 duoprism is called a 8-8 duopyramid or octagonal duopyramid. It has 64 tetragonal disphenoid cells, 128 triangular faces, 80 edges, and 16 vertices.
Skew

Related complex polygon

The regular complex polygon 28 has 16 vertices in with a real representation in matching the same vertex arrangement of the 8-8 duopyramid. It has 64 2-edges corresponding to the connecting edges of the 8-8 duopyramid, while the 16 edges connecting the two octagons are not included.
The vertices and edges makes a complete bipartite graph with each vertex from one octagon is connected to every vertex on the other.

Related polytopes

The 4-4 duoantiprism is an alternation of the 8-8 duoprism, but is not uniform. It has a highest symmetry construction of order 256 uniquely obtained as a direct alternation of the uniform 8-8 duoprism with an edge length ratio of 0.765 : 1. It has 48 cells composed of 16 square antiprisms and 32 tetrahedra. It notably occurs as a faceting of the disphenoidal 288-cell, forming part of its vertices and edges.

Vertex figure for the 4-4 duoantiprism
Also related is the bialternatosnub 4-4 duoprism, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has a highest symmetry construction of order 64, because of the alternation of square prisms and antiprisms. It has 8 cubes, 4 square antiprisms, 4 rectangular trapezoprisms, with 16 triangular prisms filling the gaps.

Vertex figure for the bialternatosnub 4-4 duoprism