3-3 duoprism

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.

3-3 duoprism

Schlegel diagram
TypeUniform duoprism
Schläfli symbol{3}×{3} = {3}2
Coxeter diagram
Cells6 triangular prisms
Faces9 squares,
6 triangles
Edges18
Vertices9
Vertex figure
Tetragonal disphenoid
Symmetry[[3,2,3]] = [6,2+,6], order 72
Dual3-3 duopyramid
Propertiesconvex, vertex-uniform, facet-transitive

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 rook's graph.

Hypervolume

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

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] This graph is also the Cayley graph of the group with generating set .

Images
Orthogonal projections
Net 3D perspective projection with 2 different rotations

Symmetry

In 5-dimensions, 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 figure. 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

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.
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 duopyramid
3-3 duopyramid
TypeUniform dual duopyramid
Schläfli symbol{3}+{3} = 2{3}
Coxeter diagram
Cells9 tetragonal disphenoids
Faces18 isosceles triangles
Edges15 (9+6)
Vertices6 (3+3)
Symmetry[[3,2,3]] = [6,2+,6], order 72
Dual3-3 duoprism
Propertiesconvex, 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

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 also

Notes
  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, S2CID 118034273
  2. Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
  3. Regular Complex Polytopes, p.110, p.114

References
  • Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
    • Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • Catalogue of Convex Polychora, section 6, George Olshevsky.
  • Apollonian Ball Packings and Stacked Polytopes Discrete & Computational Geometry, June 2016, Volume 55, Issue 4, pp 801–826
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