# 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 ${\displaystyle \mathbb {C} ^{2}}$ 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

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]

## 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 20010, 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.