3-4 duoprism

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.

Uniform 3-4 duoprisms

Schlegel diagrams
TypePrismatic uniform polychoron
Schläfli symbol{3}×{4}
Coxeter-Dynkin diagram
Cells3 square prisms,
4 triangular prisms
Faces15 squares,
4 triangles
Edges24
Vertices12
Vertex figure
Digonal disphenoid
Symmetry[3,2,4], order 48
Dual3-4 duopyramid
Propertiesconvex, vertex-uniform

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

Images

Net
3D projection with 3 different rotations
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]

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

3-4 duopyramid
3-4 duopyramid
Typeduopyramid
Schläfli symbol{3}+{4}
Coxeter-Dynkin diagram
Cells12 digonal disphenoids
Faces24 isosceles triangles
Edges19 (12+3+4)
Vertices7 (3+4)
Symmetry[3,2,4], order 48
Dual3-4 duoprism
Propertiesconvex, 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

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.
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