| 3-3 duoprism | |
| Schläfli: | × = 2 |
| Cells: | 6 triangular prisms |
| Faces: | 9 squares, 6 triangles |
| Edges: | 18 |
| Vertices: | 9 |
| Symmetry: | = [6,2<sup>+</sup>,6], order 72 |
| Property List: | convex, vertex-uniform, facet-transitive |
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope.
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 - which include 9 squares and 6 triangles. Its cell has 6 triangular prism. It has Coxeter diagram, and symmetry, order 72.
The hypervolume of a uniform 3-3 duoprism with edge length
a
The 3-3 duoprism can be represented as a graph with 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 x 3
G=\langlea,b:a3=b3=1, ab=ba\rangle\simeqC3 x C3
S=\{a,a2,b,b2\}
The minimal distance graph of a 3-3 duoprism may be ascertained by the Cartesian product of graphs between two identical both complete graphs
K3
The dual polyhedron of a 3-3 duoprism is called a 3-3 duopyramid or triangular duopyramid., page 45: "The dual of a p,q-duoprism is called a p,q-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.
The regular complex polygon 23, also 3+3 has 6 vertices in
C2
R4