3-4 duoprism

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Uniform 3-4 duoprisms
3-4 duoprism.png 140px
Schlegel diagrams
Type Prismatic uniform polychoron
Schläfli symbol {3}×{4}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Cells 3 square prisms,
4 triangular prisms
Faces 3+12 squares,
4 triangles
Edges 24
Vertices 12
Vertex figure 34-duoprism verf.png
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.


3,4 duoprism net.png
3-4 duorpism triple rotation .gif
3D projection with 3 different rotations
3-4 duoprism skew.png
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{}, CDel 3node 1.pngCDel 2.pngCDel 4node 1.png, in [math]\displaystyle{ \mathbb{C}^2 }[/math] 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, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png has a uniform 3-4 duoprism vertex figure:

Birectified penteract verf.png

3-4 duopyramid

3-4 duopyramid
Type duopyramid
Schläfli symbol {3}+{4}
Coxeter-Dynkin diagram CDel node f1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node f1.pngCDel 2x.pngCDel node f1.png
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.

3-4 duopyramid ortho.png
Orthogonal projection
3-4 duopyramid.png
Vertex-centered perspective

See also


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


  • 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