4-8 duoprism

Uniform 4-8 duoprisms Schlegel diagrams
Type	Prismatic uniform polychoron
Schläfli symbols	{4}×{8} {4}×t{4}
Coxeter diagrams
Cells	4 octagonal prisms, 8 cubes
Faces	32+8 squares, 4 octagons
Edges	64
Vertices	32
Vertex figure	Digonal disphenoid
Symmetry	[4,2,8], order 128
Dual	4-8 duopyramid
Properties	convex, vertex-uniform

In geometry of 4 dimensions, a 4-8 duoprism, a duoprism and 4-polytope resulting from the Cartesian product of a square and an octagon.

It has 12 cells (4 octagonal prisms and 8 cubes), 44 faces (40 squares and 4 octagons), 64 edges, and 32 vertices.

Images

Net

4-8 duopyramid

4-8 duopyramid
Type	duopyramid
Schläfli symbol	{4}+{8} {4}+t{4}
Coxeter-Dynkin diagram
Cells	32 digonal disphenoids
Faces	64 isosceles triangles
Edges	44 (32+4+8)
Vertices	12 (4+8)
Symmetry	[4,2,8], order 128
Dual	4-8 duoprism
Properties	convex, facet-transitive

The dual of a 4-8 duoprism is called a 4-8 duopyramid. It has 32 tetragonal disphenoid cells, 64 isosceles triangular faces, 44 edges, and 12 vertices.

Related polytopes

The 2-4 duoantiprism is an alternation of the 4-8 duoprism, but is not uniform. It has a highest symmetry construction of order 64, with 28 cells composed of 4 square antiprisms and 24 tetrahedra (8 tetragonal disphenoids and 16 digonal disphenoids). There exists a construction with uniform square antiprisms with an edge length ratio of 1 : 1.189.

Vertex figure for the 2-4 duoantiprism

Also related is the bialternatosnub 2-4 duoprism, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has a highest symmetry construction of order 32, with 4 rectangular trapezoprisms (topologically equivalent to a cube but with D_2d symmetry), 4 tetrahedra (as tetragonal disphenoids), with 8 triangular prisms (as C_2v-symmetry wedges) filling the gaps. Its vertex figure is a C_s-symmetric triangular bipyramid.

Vertex figure for the bialternatosnub 2-4 duoprism

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, 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.

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