6-8 duoprism

6-8 duoprism
Type	Prismatic uniform polychoron
Schläfli symbol	{6}×{8} {6}×t{4} t{3}×{8} t{3}×t{4}
Coxeter diagram
Cells	6 octagonal prisms, 8 hexagonal prisms
Faces	48 squares 6 octagons 8 hexagons
Edges	96
Vertices	48
Vertex figure	Digonal disphenoid
Symmetry	[6,2,8], order 192
Dual	6-8 duopyramid
Properties	convex, vertex-uniform

In geometry of 4 dimensions, a 6-8 duoprism, a duoprism and 4-polytope resulting from the Cartesian product of a hexagon and an octagon.

Images

Schlegel diagrams

Image
Center	Octagonal prism	Hexagonal prism

6-8 duopyramid

6-8 duopyramid
Type	duopyramid
Schläfli symbol	{6}+{8} {6}+t{4} t{3}+{8} t{3}+t{4}
Coxeter-Dynkin diagram
Cells	48 digonal disphenoids
Faces	96 isosceles triangles
Edges	62 (48+6+8)
Vertices	14 (6+8)
Symmetry	[6,2,8], order 192
Dual	6-8 duoprism
Properties	convex, facet-transitive

The dual of a 6-8 duoprism is called a 6-8 duopyramid. It has 48 digonal disphenoid cells, 96 isosceles triangular faces, 62 edges, and 14 vertices.

Related polytopes

The 3-4 duoantiprism is an alternation of the 6-8 duoprism, but is not uniform. It has a highest symmetry construction of order 96, with 40 cells composed of 6 square antiprisms, 8 octahedra (as triangular antiprisms), and 24 tetrahedra (as digonal disphenoids). There exists a construction with uniform square antiprisms with an edge length ratio of 1 : 1.456, and also with regular octahedra with an edge length ratio of 0.663 : 1.

Vertex figure for the 3-4 duoantiprism

Also related is the bialternatosnub 3-4 duoprism, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has a highest symmetry construction of order 48, with 6 rectangular trapezoprisms (topologically equivalent to a cube but with D_2d symmetry), 4 octahedra (as triangular antiprisms), 4 triangular prisms (both from the hexagonal prisms), with 24 triangular prisms (as C_2v-symmetry wedges) filling the gaps. Its vertex figure is a C_s-symmetric polyhedron formed by augmenting a tetrahedron on one of the square pyramid's triangular faces.

Vertex figure for the bialternatosnub 3-4 duoprism

See also

• Polytope and 4-polytope
• Convex regular 4-polytope
• Duocylinder
• Tesseract

References

• Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
• 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