# 3-4 duoprism

Uniform 3-4 duoprisms
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Schlegel diagrams
Type Prismatic uniform polychoron
Schläfli symbol {3}×{4}
Coxeter-Dynkin diagram
Cells 3 square prisms,
4 triangular prisms
Faces 3+12 squares,
4 triangles
Edges 24
Vertices 12
Vertex figure
Digonal disphenoid
Symmetry [3,2,4], order 48
Dual 3-4 duopyramid
Properties convex, vertex-uniform

In geometry of 4 dimensions, a 3-4 duoprism, the second smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of a triangle and a square.

The 3-4 duoprism exists in some of the uniform 5-polytopes in the B5 family.

## Images

 Net 3D projection with 3 different rotations Skew orthogonal projections with primary triangles and squares colored

## Related complex polygons

Stereographic projection of complex polygon, 3{}×4{} has 12 vertices and 7 3-edges, shown here with 4 red triangular 3-edges and 3 blue square 4-edges.

The quasiregular complex polytope 3{}×4{}, , in $\displaystyle{ \mathbb{C}^2 }$ has a real representation as a 3-4 duoprism in 4-dimensional space. It has 12 vertices, and 4 3-edges and 3 4-edges. Its symmetry is 3[2]4, order 12.[1]

## Related polytopes

The birectified 5-cube, has a uniform 3-4 duoprism vertex figure:

### 3-4 duopyramid

3-4 duopyramid
Type duopyramid
Schläfli symbol {3}+{4}
Coxeter-Dynkin diagram
Cells 12 digonal disphenoids
Faces 24 isosceles triangles
Edges 19 (12+3+4)
Vertices 7 (3+4)
Symmetry [3,2,4], order 48
Dual 3-4 duoprism
Properties convex, facet-transitive

The dual of a 3-4 duoprism is called a 3-4 duopyramid. It has 12 digonal disphenoid cells, 24 isosceles triangular faces, 12 edges, and 7 vertices.

 Orthogonal projection Vertex-centered perspective

## Notes

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

## 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.