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.

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 [math]\displaystyle{ \mathbb{C}^2 }[/math] 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
Skew	[6]	[12]

Related complex polygon

Orthographic projection

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

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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