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3-4 duoprism

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Uniform 3-4 duoprisms

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

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Net

3D projection with 3 different rotations

Skew orthogonal projections with primary triangles and squares colored
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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 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]

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

3-4 duopyramid

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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

See also

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Notes

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

References

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  • 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-Strauss, 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.
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