Edge-contracted icosahedron

Short description: Convex polyhedron with 18 triangular faces

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
Net

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

## Contents

* 1 Construction
* 2 Related polytopes
* 3 In chemistry
* 4 Related polyhedra
* 5 References
* 6 External links

## 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 [B
11H
11]2−. [3]

Ball-and-stick model of the
closo-undecaborate ion, [B
11H
11]2− |
closo-boranate [B
11H
11]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. ↑ "Convex Triamond Regular Polyhedra". http://www.interocitors.com/polyhedra/Triamonds/.
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, 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

* v
* t
* e

Polyhedra

Listed by number of faces

1–10 faces|

* Monohedron
* Dihedron
* Trihedron
* Tetrahedron
* Pentahedron
* Hexahedron
* Heptahedron
* Octahedron
* Enneahedron
* Decahedron

11–20 faces|

* Hendecahedron
* Dodecahedron
* Tridecahedron
* Tetradecahedron
* Pentadecahedron
* Hexadecahedron
* Heptadecahedron
* Octadecahedron
* Enneadecahedron
* Icosahedron

Others|

* Triacontahedron (30)
* Hexecontahedron (60)
* Enneacontahedron (90)
* Hectotriadiohedron
* Apeirohedron

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Edge-contracted icosahedron. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Edge-contracted_icosahedron&oldid=227156"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template