3 3 duoprism

Symmetry

In 5-dimensions, the some uniform 5-polytopes have 3-3 duoprism vertex figures, some with unequal edge-lengths and therefore lower symmetry:

The birectified 16-cell honeycomb also has a 3-3 duoprism vertex figures. There are three constructions for the honeycomb with two lower symmetries.

Related complex polygons

The regular complex polytope ₃{4}₂, , in C 2 has a real representation as a 3-3 duoprism in 4-dimensional space. ₃{4}₂ has 9 vertices, and 6 3-edges. Its symmetry is ₃[4]₂, order 18. It also has a lower symmetry construction, , or ₃{}×₃{}, with symmetry ₃[2]₃, order 9. This is the symmetry if the red and blue 3-edges are considered distinct.

3-3 duopyramid

The dual of a 3-3 duoprism is called a 3-3 duopyramid. It has 9 tetragonal disphenoid cells, 18 triangular faces, 15 edges, and 6 vertices.

It can be seen in orthogonal projection as a 6-gon circle of vertices, and edges connecting all pairs, just like a 5-simplex seen in projection.

Related complex polygon

The regular complex polygon ₂{4}₃ has 6 vertices in C 2 with a real represention in R 4 matching the same vertex arrangement of the 3-3 duopyramid. It has 9 2-edges corresponding to the connecting edges of the 3-3 duopyramid, while the 6 edges connecting the two triangles are not included. It can be seen in a hexagonal projection with 3 sets of colored edges. This arrangement of vertices and edges makes a complete bipartite graph with each vertex from one triangle is connected to every vertex on the other. It is also called a Thomsen graph or 4-cage.