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Edge-contracted icosahedron
Type	Octadecahedron
Faces	18 triangles
Edges	27
Vertices	11
Vertex configuration	2 (3) 8 (3) 1 (3)
Symmetry group	C2v, , (*22), order 4
Properties	Convex, deltahedron
Net

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

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 three coplanar triangles are considered a single face (called a triamond), 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 variant topologically equivalent to the sphenocorona 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, 22 edges, and 12 equilateral triangular faces and 2 trapezoid faces.

In chemistry

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

Ball-and-stick model of the closo-undecaborate ion, [B11H11]

closo-boranate [B11H11]

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. "Convex Triamond Regular Polyhedra".
2. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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

External links

• The Convex Deltahedra, And the Allowance of Coplanar Faces

• Polyhedra