| 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. 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
3 x 3
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.
The hypervolume of a uniform 3-3 duoprism with edge length
a
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 x 3
G=\langlea,b:a3=b3=1, ab=ba\rangle\simeqC3 x C3
S=\{a,a2,b,b2\}
In 5-dimensions, some uniform 5-polytopes have 3-3 duoprism vertex figures, some with unequal edge-lengths and therefore lower symmetry:
| Symmetry | , order 72 | [3,2], order 12 | |||
|---|---|---|---|---|---|
| Coxeter diagram | |||||
| Schlegel diagram | |||||
| Name | t03α5 | t03γ5 | t03β5 | ||
The regular complex polytope 32,, in
C2
| bgcolor=#e7dcc3 colspan=2 | 3-3 duopyramid | |
|---|---|---|
| Type | Uniform dual duopyramid | |
| Schläfli symbol | + = 2 | |
| Coxeter diagram | ||
| Cells | 9 tetragonal disphenoids | |
| Faces | 18 isosceles triangles | |
| Edges | 15 (9+6) | |
| Vertices | 6 (3+3) | |
| Symmetry | = [6,2<sup>+</sup>,6], order 72 | |
| Dual | 3-3 duoprism | |
| Properties | convex, vertex-uniform, facet-transitive |
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.
orthogonal projection
The regular complex polygon 23, also 3+3 has 6 vertices in
C2
R4