# Edge-contracted icosahedron

Edge-contracted icosahedron
Faces 18 triangles
Edges 27
Vertices 11
Symmetry C2v, [2], (*22), order 4
Vertex configuration 2 (34)
8 (35)
1 (36)
Properties Convex, deltahedron
Net

In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices with C2v symmetry, order 4.

## Construction

It can be constructed from the regular icosahedron, with one edge contraction, removing one vertex, 3 edges, and 2 faces. This contraction distorts the circumscribed sphere original vertices. With all equilateral triangle faces, it has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid.

If the sets of coplanar triangles are considered a single face (called a triamond[1]), it has 10 vertices, 22 edges, and 14 faces, 12 triangles and 2 triamonds .

It may also be described as having a hybrid square-pentagonal antiprismatic core (an antiprismatic core with one square base and one pentagonal base); each base is then augmented with a pyramid.

## Related polytopes

The dissected regular icosahedron is a name for this polytope with the two sets of 3 coplanar faces as trapezoids. This is the vertex figure of a 4D polytope, grand antiprism. It has 10 vertices, 23 edges, and 11 equilateral triangular faces and 2 trapezoid faces.[2]

## In chemistry

In chemistry, this polyhedron is most commonly called the octadecahedron, for 18 triangular faces, and represents the closo-boranate [B11H11]2−. [3]

 closo-boranate [B11H11]2− Net

## Related polyhedra

The elongated octahedron is similar to the edge-contracted icosahedron, but instead of only one edge contracted, two opposite edges are contracted.

## References

1. ^ http://www.interocitors.com/polyhedra/Triamonds/
2. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26) The Grand Antiprism
3. ^ Holleman, Arnold Frederik; Wiberg, Egon (2001), Wiberg, Nils, ed., Inorganic Chemistry, translated by Eagleson, Mary; Brewer, William, San Diego/Berlin: Academic Press/De Gruyter, p. 1165, ISBN 0-12-352651-5