10-10 duoprism

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 ${\displaystyle \mathbb {C} ^{2}}$ 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
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 ${\displaystyle \mathbb {C} ^{2}}$ with a real representation in ${\displaystyle \mathbb {R} ^{4}}$ 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

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

• 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.

• 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