3-4 duoprism explained

bgcolor=#e7dcc3 colspan=2	Uniform 3-4 duoprisms Schlegel diagrams
Type	Prismatic uniform polychoron
Schläfli symbol	×
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.

Related complex polygons

The quasiregular complex polytope ₃×₄,, in

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

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

3-4 duopyramid

bgcolor=#e7dcc3 colspan=2	3-4 duopyramid
Type	duopyramid
Schläfli symbol	+
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.

See also

• Polytope and polychoron
• Convex regular polychoron
• Duocylinder
• Tesseract

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, (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, (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

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

Notes and References

1. [Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]