3-3 duoprism

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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.

Hypervolume[edit]

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[edit]

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]

Images[edit]

Orthogonal projections
3-3 duoprism ortho-dih3.png 3-3 duoprism ortho-skew.png 3-3 duoprism ortho-Dih3.png 3-3 duoprism ortho square.png
3,3 duoprism net.png 3-3 duopyramid.png Triangular Duoprism YW and ZW Rotations.gif
Net Vertex-centered perspective 3D perspective projection with 2 different rotations

Symmetry[edit]

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
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Schlegel
diagram
Birectified hexateron verf.png Runcinated 5-simplex verf.png Runcinated penteract verf.png Runcinated pentacross verf.png
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
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png CDel node.pngCDel 3.pngCDel node.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
Skew
orthogonal
projection
Birectified 16-cell honeycomb verf.png Birectified 16-cell honeycomb verf2.png Birectified 16-cell honeycomb verf3.png

Related complex polygons[edit]

The regular complex polytope 3{4}2, CDel 3node 1.pngCDel 4.pngCDel node.png, 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, CDel 3node 1.pngCDel 2.pngCDel 3node 1.png, 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]

Complex polygon 3-4-2-stereographic2.png
Perspective projection
3-generalized-2-cube.svg
Orthogonal projection with coinciding central vertices
3-generalized-2-cube skew.svg
Orthogonal projection, offset view to avoid overlapping elements.

Related polytopes[edit]

k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6 =E6+ =E6++
Coxeter
diagram
CDel nodes.pngCDel 3ab.pngCDel nodes 11.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry [[32,2,-1]] [[32,2,0]] [[32,2,1]] [[32,2,2]] [[32,2,3]]
Order 72 1440 103,680
Graph 3-3 duoprism ortho-skew.png 5-simplex t2.svg Up 1 22 t0 E6.svg
Name −122 022 122 222 322

3-3 duopyramid[edit]

3-3 duopyramid
Type Uniform dual duopyramid
Schläfli symbol {3}+{3} = 2{3}
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node f1.pngCDel 3.pngCDel node.png
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.

3-3 duopyramid ortho.png
orthogonal projection

Related complex polygon[edit]

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]

Complex polygon 2-4-3-bipartite graph.png
The 2{4}3 with 6 vertices in blue and red connected by 9 2-edges as a complete bipartite graph.
Complex polygon 2-4-3.png
It has 3 sets of 3 edges, seen here with colors.

See also[edit]

Notes[edit]

  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

References[edit]

External links[edit]