3-3 duoprism

3-3 duoprisms Schlegel diagram
Type	Uniform duoprism
Schläfli symbol	{3}×{3} = {3}2
Coxeter diagram
Cells	6 triangular prisms
Faces	9 squares, 6 triangles
Edges	18
Vertices	9
Vertex figure	Tetragonal disphenoid
Symmetry	[[3,2,3]] = [6,2+,6], order 72
Dual	3-3 duopyramid
Properties	convex, vertex-uniform, facet-transitive

In geometry of 4 dimensions, a 3-3 duoprism, the smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of two triangles.

It has 9 vertices, 18 edges, 48 faces (9 squares, and 6 triangles), in 6 triangular prism cells. It has Coxeter diagram , and symmetry [[3,2,3]], order 72.

Images

2D orthogonal projection
Net		Vertex-centered perspective

Symmetry

In 5-dimensions, the 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 figures. 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

Related polytopes

k₂₂ figures in n dimensions

Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	2A2	A5	E6	=E6+	=E6++
Coxeter diagram
Symmetry	[[3<sup>2,2,-1</sup>]]	[[3<sup>2,2,0</sup>]]	[[3<sup>2,2,1</sup>]]	[[3<sup>2,2,2</sup>]]	[[3<sup>2,2,3</sup>]]
Order	72	1440	103,680	∞
Graph				∞	∞
Name	−122	022	122	222	322

3-3 duopyramid

3-3 duopyramid
Type	Uniform dual duopyramid
Schläfli symbol	{3}+{3} = 2{3}
Coxeter diagram
Cells	9 tetragonal disphenoids
Faces	18 isosceles triangles
Edges	15 (9+6)
Vertices	6 (3+3)
Symmetry	[[3,2,3]] = [6,2+,6], order 72
Dual	3-3 duoprism
Properties	convex, vertex-uniform, facet-transitive

The dual of a 3-3 duoprism is called a 3-3 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

See also

• 3-4 duoprism
• Tesseract (4-4 duoprism)
• 5-5 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
• Olshevsky, George, Duoprism at Glossary for Hyperspace.
  • 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