8-8 duoprism

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Uniform 8-8 duoprism
8-8 duoprism.png
Schlegel diagram
Type Uniform duoprism
Schläfli symbol {8}×{8} = {8}2
Coxeter diagrams CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Cells 16 octagonal prisms
Faces 64 squares,
16 octagons
Edges 128
Vertices 64
Vertex figure Tetragonal disphenoid
Symmetry [[8,2,8]] = [16,2+,16], order 512
Dual 8-8 duopyramid
Properties convex, 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 CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel node.png, and symmetry [[8,2,8]], order 512.

Images[edit]

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
8,8 duoprism net0.png 8,8 duoprism net.png 8,8 duoprism net3.png 8,8 duoprism net2.png
{8}2 {8}×{8} t{4}2 t{4}×t{4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png

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 ortho-Dih8.png 4-cube t0.svg 7-simplex t06.svg
8-8 duoprism
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
4-4 duoprism
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Hexicated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Ammann-Beenker tiling example.png Octagonal star2.png Star polygon 8-3.svg
Ammann–Beenker tiling {8/3} octagrams
8-8 duoprism ortho-3.png
8-8 duoprism

Related complex polygons[edit]

Orthogonal projection shows 8 red and 8 blue outlined 8-edges

The regular complex polytope 8{4}2, CDel 8node 1.pngCDel 4.pngCDel node.png, 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, CDel 8node 1.pngCDel 2.pngCDel 8node 1.png, 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[edit]

8-8 duopyramid
Type Uniform dual duopyramid
Schläfli symbol {8}+{8} = 2{8}
Coxeter diagrams CDel node f1.pngCDel 8.pngCDel node.pngCDel 2x.pngCDel node f1.pngCDel 8.pngCDel node.png
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 8.pngCDel node.png
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Cells 64 tetragonal disphenoids
Faces 128 isosceles triangles
Edges 80 (64+16)
Vertices 16 (8+8)
Symmetry [[8,2,8]] = [16,2+,16], order 512
Dual 8-8 duoprism
Properties convex, 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
8-8 duopyramid ortho.png 8-8-duopyramid.svg
Skew [16]

Related complex polygon[edit]

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[edit]

Notes[edit]

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

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-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[edit]