8-8 duoprism

Uniform 8-8 duoprism

Schlegel diagram
TypeUniform duoprism
Schläfli symbol{8}×{8} = {8}2
Coxeter diagrams

Cells16 octagonal prisms
Faces64 squares,
16 octagons
Edges128
Vertices64
Vertex figureTetragonal disphenoid
Symmetry[[8,2,8]] = [16,2+,16], order 512
Dual8-8 duopyramid
Propertiesconvex, vertex-uniform, facet-transitive

In geometry of 4 dimensions, a 8-8 duoprism or octagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two octagons.

It has 64 vertices, 128 edges, 80 faces (64 squares, and 16 octagons), in 16 octagonal prism cells. It has Coxeter diagram , and symmetry [[8,2,8]], order 512.

Images

A perspective view of half of the octagonal prisms along one direction, alternately colored.

The uniform 8-8 duoprism can be constructed from [8]×[8] or [4]×[4] symmetry, order 256 or 64, with extended symmetry doubling these with a 2-fold rotation that maps the two orientations of prisms together. These can be expressed by 4 permutations of uniform coloring of the octagonal prism cells.

Uniform colored nets
[[8,2,8]], order 512 [8,2,8], order 256 [[4,2,4]], order 128 [4,2,4], order 64
{8}2{8}×{8} t{4}2t{4}×t{4}

Seen in a skew 2D orthogonal projection, it has the same vertex positions as the hexicated 7-simplex, except for a center vertex. The projected rhombi and squares are also shown in the Ammann–Beenker tiling.

8-8 duoprism
 4-4 duoprism
 Hexicated 7-simplex
Ammann–Beenker tiling {8/3} octagrams
8-8 duoprism
Orthogonal projection shows 8 red and 8 blue outlined 8-edges

The regular complex polytope 8{4}2, , in has a real representation as a 8-8 duoprism in 4-dimensional space. 8{4}2 has 64 vertices, and 16 8-edges. Its symmetry is 8[4]2, order 128.

It also has a lower symmetry construction, , or 8{}×8{}, with symmetry 8[2]8, order 64. This is the symmetry if the red and blue 8-edges are considered distinct.[1]

8-8 duopyramid

8-8 duopyramid
TypeUniform dual duopyramid
Schläfli symbol{8}+{8} = 2{8}
Coxeter diagrams

Cells64 tetragonal disphenoids
Faces128 isosceles triangles
Edges80 (64+16)
Vertices16 (8+8)
Symmetry[[8,2,8]] = [16,2+,16], order 512
Dual8-8 duoprism
Propertiesconvex, vertex-uniform, facet-transitive

The dual of a 8-8 duoprism is called a 8-8 duopyramid or octagonal duopyramid. It has 64 tetragonal disphenoid cells, 128 triangular faces, 80 edges, and 16 vertices.

orthogonal projections
Skew [16]
Orthographic projection

The regular complex polygon 2{4}8 has 16 vertices in with a real represention in matching the same vertex arrangement of the 8-8 duopyramid. It has 64 2-edges corresponding to the connecting edges of the 8-8 duopyramid, while the 16 edges connecting the two octagons are not included.

The vertices and edges makes a complete bipartite graph with each vertex from one octagon is connected to every vertex on the other.[2]

See also

Notes

  1. ↑ Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
  2. ↑ Regular Complex Polytopes, p.114

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.
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