6-6 duoprism

Uniform 6-6 duoprism

Schlegel diagram
TypeUniform duoprism
Schläfli symbol{6}×{6} = {6}2
Coxeter diagrams
Cells12 hexagonal prisms
Faces36 squares,
12 hexagons
Edges72
Vertices36
Vertex figureTetragonal disphenoid
Symmetry[[6,2,6]] = [12,2+,12], order 288
Dual6-6 duopyramid
Propertiesconvex, vertex-uniform, facet-transitive

In geometry of 4 dimensions, a 6-6 duoprism or hexagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two hexagons.

It has 36 vertices, 72 edges, 48 faces (36 squares, and 12 hexagons), in 12 hexagonal prism cells. It has Coxeter diagram , and symmetry [[6,2,6]], order 288.

Images


Net

Seen in a skew 2D orthogonal projection, it contains the projected rhombi of the rhombic tiling.

6-6 duoprism Rhombic tiling
6-6 duoprism 6-6 duoprism
Orthogonal projection shows 6 red and 6 blue outlined 6-edges

The regular complex polytope 6{4}2, , in has a real representation as a 6-6 duoprism in 4-dimensional space. 6{4}2 has 36 vertices, and 12 6-edges. Its symmetry is 6[4]2, order 72. It also has a lower symmetry construction, , or 6{}×6{}, with symmetry 6[2]6, order 36. This is the symmetry if the red and blue 6-edges are considered distinct.[1]

6-6 duopyramid

6-6 duopyramid
TypeUniform dual duopyramid
Schläfli symbol{6}+{6} = 2{6}
Coxeter diagrams
Cells36 tetragonal disphenoids
Faces72 isosceles triangles
Edges48 (36+12)
Vertices12 (6+6)
Symmetry[[6,2,6]] = [12,2+,12], order 288
Dual6-6 duoprism
Propertiesconvex, vertex-uniform,
facet-transitive

The dual of a 6-6 duoprism is called a 6-6 duopyramid or hexagonal duopyramid. It has 36 tetragonal disphenoid cells, 72 triangular faces, 48 edges, and 12 vertices.

It can be seen in orthogonal projection:

Skew [6] [12]
Orthographic projection

The regular complex polygon 2{4}6 or has 12 vertices in with a real representation in matching the same vertex arrangement of the 6-6 duopyramid. It has 36 2-edges corresponding to the connecting edges of the 6-6 duopyramid, while the 12 edges connecting the two hexagons are not included.

The vertices and edges makes a complete bipartite graph with each vertex from one pentagon is connected to every vertex on the other.[2]

See also

Notes

  1. ↑ Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
  2. ↑ Regular Complex Polytopes, 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.
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