3-3 duoprism
3-3 duoprism.png
Schlegel diagram
Type Uniform duoprism
Schläfli symbol {3}×{3} = {3}2
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 6 triangular prisms
Faces 9 squares,
6 triangles
Edges 18
Vertices 9
Vertex figure 33-duoprism verf.png
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 CDel branch 10.pngCDel 2.pngCDel branch 10.png, and symmetry [[3,2,3]], order 72. Its vertices and edges form a rook's graph.

HypervolumeEdit

The hypervolume of a uniform 3-3 duoprism, with edge length a, is  . This is the square of the area of an equilateral triangle,  .

GraphEdit

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.[1]

ImagesEdit

Orthogonal projections
       
     
Net Vertex-centered perspective 3D perspective projection with 2 different rotations

SymmetryEdit

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 [3], order 6
Coxeter
diagram
                        
Skew
orthogonal
projection
     

Related complex polygonsEdit

The regular complex polytope 3{4}2,    , in   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.[2]

 
Perspective projection
 
Orthogonal projection with coinciding central vertices
 
Orthogonal projection, offset view to avoid overlapping elements.

Related polytopesEdit

k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6  =E6+  =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 duopyramidEdit

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 polygonEdit

The regular complex polygon 2{4}3 has 6 vertices in   with a real representation in   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.[3]

 
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.

See alsoEdit

NotesEdit

  1. ^ Makhnev, A. A.; Minakova, I. M. (January 2004), "On automorphisms of strongly regular graphs with parameters  ,  ", Discrete Mathematics and Applications, 14 (2), doi:10.1515/156939204872374, MR 2069991
  2. ^ Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
  3. ^ Regular Complex Polytopes, p.110, p.114

ReferencesEdit

External linksEdit