3-4_duoprism Knowpia

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 edit

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

Related complex polygons edit

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 $\mathbb {C} ^{2}$ 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 edit

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

3-4 duopyramid edit

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 edit

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

References edit

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.

External links edit

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

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

[1]

Summary