# 3-4 duoprism

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Uniform 3-4 duoprisms 140px 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

The quasiregular complex polytope _{3}{}×_{4}{}, , 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, 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 |

## See also

- Polytope and polychoron
- Convex regular polychoron
- Duocylinder
- Tesseract

## Notes

- ↑ 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.

- Coxeter, H. S. M.
- 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

- N.W. Johnson:
- 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

Original source: https://en.wikipedia.org/wiki/3-4 duoprism.
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