3-3 duoprism


In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the duoprisms.
It has 9 vertices, 18 edges, 15 faces, in 6 triangular prism cells. It has Coxeter diagram, and symmetry, order 72. Its vertices and edges form a rook's graph.

Hypervolume

The hypervolume of a 3-3 duoprism, with edge length a, is. This is the square of the area of an equilateral triangle,.

Graph

The graph of vertices and edges of the 3-3 duoprism has 9 vertices and 18 edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique square. It is a toroidal graph, a locally linear graph, a strongly regular graph with parameters, the rook's graph, and the Paley graph of order 9.

Images

NetVertex-centered perspective3D perspective projection with 2 different rotations

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.
Symmetry, order 36, order 12, order 6
Coxeter
diagram
Skew
orthogonal
projection

Related complex polygons

The regular complex polytope 32,, in has a real representation as a 3-3 duoprism in 4-dimensional space. 32 has 9 vertices, and 6 3-edges. Its symmetry is 32, order 18. It also has a lower symmetry construction,, or 3×3, with symmetry 33, order 9. This is the symmetry if the red and blue 3-edges are considered distinct.

Perspective projection
Orthogonal projection with coinciding central vertices
Orthogonal projection, offset view to avoid overlapping elements.

Related polytopes

3-3 duopyramid

The dual of a 3-3 duoprism is called a 3-3 duopyramid or triangular 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 23 has 6 vertices in with a real representation in 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.

The 23 with 6 vertices in blue and red connected by 9 2-edges as a complete bipartite graph.
It has 3 sets of 3 edges, seen here with colors.