# 3-3 duoprism

3-3 duoprism Schlegel diagram
Type Uniform duoprism
Schläfli symbol {3}×{3} = {3}2
Coxeter diagram       Cells 6 triangular prisms
Faces 9 squares,
6 triangles
Edges 18
Vertices 9
Vertex figure Tetragonal disphenoid
Symmetry [[3,2,3]] = [6,2+,6], order 72
Dual 3-3 duopyramid
Properties 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 [[3,2,3]], order 72. Its vertices and edges form a $3\times 3$ rook's graph.

## Hypervolume

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, $A={{\sqrt {3}} \over 4}a^{2}$ .

## 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 $3\times 3$  rook's graph, and the Paley graph of order 9.

## Images

Net Vertex-centered perspective 3D 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:

Symmetry [[3,2,3]], order 72 [3,2], order 12
Coxeter
diagram

Schlegel
diagram

Name t2α5 t03α5 t03γ5 t03β5

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 [3,2,3], order 36 [3,2], order 12 , order 6
Coxeter
diagram

Skew
orthogonal
projection

## 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 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

k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6 ${\tilde {E}}_{6}$ =E6+ ${\bar {T}}_{7}$ =E6++
Coxeter
diagram

Symmetry [[32,2,-1]] [[32,2,0]] [[32,2,1]] [[32,2,2]] [[32,2,3]]
Order 72 1440 103,680
Graph
Name −122 022 122 222 322

### 3-3 duopyramid

3-3 duopyramid
Type Uniform dual duopyramid
Schläfli symbol {3}+{3} = 2{3}
Coxeter diagram
Cells 9 tetragonal disphenoids
Faces 18 isosceles triangles
Edges 15 (9+6)
Vertices 6 (3+3)
Symmetry [[3,2,3]] = [6,2+,6], order 72
Dual 3-3 duoprism
Properties convex, vertex-uniform, facet-transitive

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.

orthogonal projection

#### Related complex polygon

The regular complex polygon 2{4}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. The 2{4}3 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.