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 (9 squares, and 6 triangles), in 6 triangular prism cells. It has Coxeter diagram, and symmetry [3,2,3], order 72. Its vertices and edges form a $$3\times 3$$ rook's graph.

Description
The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons. In the case of 3-3 duoprism is the simplest among them, and it can be constructed using Cartesian product of two triangles. The resulting duoprism has 9 vertices, 18 edges, and 15 faces&mdash;which include 9 squares and 6 triangles. Its cell has 6 triangular prism.

The hypervolume of a uniform 3-3 duoprism with edge length $$ a $$ is $$ V_4 = {3\over 16}a^4.$$ This is the square of the area of an equilateral triangle, $$ A = {\sqrt3\over 4}a^2. $$

The 3-3 duoprism can be represented as a graph, which has the same number of vertices and 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 (9,4,1,2), the $$3\times 3$$ rook's graph, and the Paley graph of order 9. This graph is also the Cayley graph of the group $$G=\langle a,b:a^3=b^3=1,\ ab=ba\rangle\simeq C_3\times C_3$$ with generating set $$S=\{a,a^2,b,b^2\}$$.

Symmetry
In 5-dimensions, 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 figure. There are three constructions for the honeycomb with two lower symmetries.

Related complex polygons
The regular complex polytope 3{4}2,, in $$\mathbb{C}^2$$ 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 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.
 * 3-3 duopyramid ortho.png orthogonal projection

Related complex polygon
The regular complex polygon 2{4}3, also 3{ }+3{ } has 6 vertices in $$\mathbb{C}^2$$ with a real representation in $$\mathbb{R}^4$$ 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.