10-10 duoprism

Uniform 10-10 duoprism Schlegel diagram
Type	Uniform duoprism
Schläfli symbol	{10}×{10} = {10}2
Coxeter diagrams
Cells	25 decagonal prisms
Faces	100 squares, 20 decagons
Edges	200
Vertices	100
Vertex figure	Tetragonal disphenoid
Symmetry	[[10,2,10]] = [20,2+,20], order 800
Dual	10-10 duopyramid
Properties	convex, vertex-uniform, Facet-transitive

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.

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]

Related complex polygons

Orthogonal projection shows 10 red and 10 blue outlined 10-edges

The regular complex polytope ₁₀{4}₂, , in [math]\displaystyle{ \mathbb{C}^2 }[/math] 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.

10-10 duopyramid

10-10 duopyramid
Type	Uniform dual duopyramid
Schläfli symbol	{10}+{10} = 2{10}
Coxeter diagrams
Cells	100 tetragonal disphenoids
Faces	200 isosceles triangles
Edges	120 (100+20)
Vertices	20 (10+10)
Symmetry	[[10,2,10]] = [20,2+,20], order 800
Dual	10-10 duoprism
Properties	convex, 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

Related complex polygon

Orthographic projection

The regular complex polygon ₂{4}₁₀ has 20 vertices in [math]\displaystyle{ \mathbb{C}^2 }[/math] with a real representation in [math]\displaystyle{ \mathbb{R}^4 }[/math] 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.

Related polytopes

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

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

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.

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