6-6 duoprism

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.

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

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

Related complex polygons

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

The regular complex polytope ₆{4}₂, , in ${\displaystyle \mathbb {C} ^{2}}$ 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:

Skew	[6]	[12]

Related complex polygon

Orthographic projection

The regular complex polygon ₂{4}₆ or has 12 vertices in ${\displaystyle \mathbb {C} ^{2}}$ with a real representation in ${\displaystyle \mathbb {R} ^{4}}$ 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