Edge-contracted icosahedron

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

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

Ball-and-stick model of the closo-undecaborate ion, [B11H11]2−

closo-boranate [B11H11]2−

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