6-6 duoprism

Uniform 6-6 duoprism

Schlegel diagram

Type | Uniform duoprism
Schläfli symbol | {6}×{6} = {6}2
Coxeter diagrams |

Cells | 12 hexagonal prisms
Faces | 36 squares,
12 hexagons
Edges | 72
Vertices | 36
Vertex figure | Tetragonal disphenoid
Symmetry | [[6,2,6]] = [12,2+,12], order 288
Dual | 6-6 duopyramid
Properties | convex, 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.

## Contents

* 1 Images
* 2 Related complex polygons
* 3 6-6 duopyramid
* 3.1 Related complex polygon
* 4 Related polytopes
* 5 See also
* 6 Notes
* 7 References
* 8 External links

## Images

Net

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

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6-6 duoprism  | Rhombic tiling

6-6 duoprism  | 6-6 duoprism

## Related complex polygons

Orthogonal projection shows 6 red and 6 blue outlined 6-edges

The regular complex polytope 6{4}2, , in ℂ2 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

Type | Uniform dual duopyramid
Schläfli symbol | {6}+{6} = 2{6}
Coxeter diagrams |

Cells | 36 tetragonal disphenoids
Faces | 72 isosceles triangles
Edges | 48 (36+12)
Vertices | 12 (6+6)
Symmetry | [[6,2,6]] = [12,2+,12], order 288
Dual | 6-6 duoprism
Properties | convex, 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:

| 120px |

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Skew  | [6]  | [12]

### Related complex polygon

Orthographic projection

The regular complex polygon 2{4}6 or  has 12 vertices in ℂ2 with a real representation in ℝ4 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]

## Related polytopes

The 3-3 duoantiprism is an alternation of the 6-6 duoprism, but is not uniform. It has a highest symmetry construction of order 144 uniquely obtained as a direct alternation of the uniform 6-6 duoprism with an edge length ratio of 0.816 : 1. It has 30 cells composed of 12 octahedra (as triangular antiprisms) and 18 tetrahedra (as tetragonal disphenoids). The vertex figure is a gyrobifastigium, which has a regular-faced variant that is not isogonal. It is also the convex hull of two uniform 3-3 duoprisms in opposite positions.

Vertex figure for the 3-3 duoantiprism

## See also

* 3-3 duoprism
* 3-4 duoprism
* 5-5 duoprism
* Tesseract (4-4 duoprism)
* Convex regular 4-polytope
* Duocylinder

## 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.

## External links

* The Fourth Dimension Simply Explained--describes duoprisms as "double prisms" and duocylinders as "double cylinders"
* Polygloss \- glossary of higher-dimensional terms
* Exploring Hyperspace with the Geometric Product

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