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 , order 72. Its vertices and edges form a rook's graph. Hypervolume The hypervolume of a uniform 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 (9,4,1,2), the rook's graph, and the Paley graph of order 9. This graph is also the Cayley graph of the group with generating set