10-10 duoprism

In geometry of 4 dimensions, a 10-10 duoprism or decagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two decagons.

Uniform 10-10 duoprism


Schlegel diagram
TypeUniform duoprism
Schläfli symbol{10}×{10} = {10}2
Coxeter diagrams

Cells25 decagonal prisms
Faces100 squares,
20 decagons
Edges200
Vertices100
Vertex figureTetragonal disphenoid
Symmetry[[10,2,10]] = [20,2+,20], order 800
Dual10-10 duopyramid
Propertiesconvex, vertex-uniform, Facet-transitive

It has 100 vertices, 200 edges, 120 faces (100 squares, and 20 decagons), in 20 decagonal prism cells. It has Coxeter diagram , and symmetry [[10,2,10]], order 800.

Images

The uniform 10-10 duoprism can be constructed from [10]×[10] or [5]×[5] symmetry, order 400 or 100, with extended symmetry doubling these with a 2-fold rotation that maps the two orientations of prisms together.

2D orthogonal projection Net
[10] [20]
Orthogonal projection shows 10 red and 10 blue outlined 10-edges

The regular complex polytope 10{4}2, , in has a real representation as a 10-10 duoprism in 4-dimensional space. 10{4}2 has 100 vertices, and 20 10-edges. Its symmetry is 10[4]2, order 200.

It also has a lower symmetry construction, , or 10{}×10{}, with symmetry 10[2]10, order 100. This is the symmetry if the red and blue 10-edges are considered distinct.[1]

10-10 duopyramid
10-10 duopyramid
TypeUniform dual duopyramid
Schläfli symbol{10}+{10} = 2{10}
Coxeter diagrams

Cells100 tetragonal disphenoids
Faces200 isosceles triangles
Edges120 (100+20)
Vertices20 (10+10)
Symmetry[[10,2,10]] = [20,2+,20], order 800
Dual10-10 duoprism
Propertiesconvex, vertex-uniform, Facet-transitive

The dual of a 10-10 duoprism is called a 10-10 duopyramid or decagonal duopyramid. It has 100 tetragonal disphenoid cells, 200 triangular faces, 120 edges, and 20 vertices.


Orthogonal projection

Orthographic projection

The regular complex polygon 2{4}10 has 20 vertices in with a real representation in matching the same vertex arrangement of the 10-10 duopyramid. It has 100 2-edges corresponding to the connecting edges of the 10-10 duopyramid, while the 20 edges connecting the two decagons are not included.

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

The 5-5 duoantiprism is an alternation of the 10-10 duoprism, but is not uniform. It has a highest symmetry construction of order 400 uniquely obtained as a direct alternation of the uniform 10-10 duoprism with an edge length ratio of 0.743 : 1. It has 70 cells composed of 20 pentagonal antiprisms and 50 tetrahedra (as tetragonal disphenoids).


Vertex figure for the 5-5 duoantiprism

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 978-0-486-40919-1 (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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