8-8 duoprism

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.

Uniform 8-8 duoprism Schlegel diagram
Type	Uniform duoprism
Schläfli symbol	{8}×{8} = {8}^2
Coxeter diagrams
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

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}^2	t{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

Related complex polygons

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

The regular complex polytope ₈{4}₂, , in ${\displaystyle \mathbb {C} ^{2}}$ has a real representation as an 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
Type	Uniform dual duopyramid
Schläfli symbol	{8}+{8} = 2{8}
Coxeter diagrams
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

Skew	[16]

Related complex polygon

Orthographic projection

The regular complex polygon ₂{4}₈ has 16 vertices in ${\displaystyle \mathbb {C} ^{2}}$ with a real representation in ${\displaystyle \mathbb {R} ^{4}}$ 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]

Related polytopes

The 4-4 duoantiprism is an alternation of the 8-8 duoprism, but is not uniform. It has a highest symmetry construction of order 256 uniquely obtained as a direct alternation of the uniform 8-8 duoprism with an edge length ratio of 0.765 : 1. It has 48 cells composed of 16 square antiprisms and 32 tetrahedra (as tetragonal disphenoids). It notably occurs as a faceting of the disphenoidal 288-cell, forming part of its vertices and edges.

Vertex figure for the 4-4 duoantiprism

Also related is the bialternatosnub 4-4 duoprism, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has a highest symmetry construction of order 64, because of the alternation of square prisms and antiprisms. It has 8 cubes (as square prisms), 4 square antiprisms, 4 rectangular trapezoprisms (topologically equivalent to a cube but with D_2d symmetry), with 16 triangular prisms (as C_sv-symmetry wedges) filling the gaps.

Vertex figure for the bialternatosnub 4-4 duoprism

See also

• 3-3 duoprism
• 3-4 duoprism
• 5-5 duoprism
• Tesseract (4-4 duoprism)
• Convex regular 4-polytope
• Duocylinder

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.

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