In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27

edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

, and 11 vertices.

Construction

It can be constructed from the regular

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

, 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 In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

. 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 Polyiamond-1-1

and 2 triamonds Polyiamond-3-1

. It may also be described as having a hybrid square-

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

al antiprismatic core (an antiprismatic core with one square base and one pentagonal base); each base is then

augmented Augment or augmentation may refer to: Language * Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages *Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns ...

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

chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

, this polyhedron is most commonly called the octadecahedron, for 18 triangular faces, and represents the closo-boranate .

Related polyhedra

The

elongated octahedron In geometry, an elongated octahedron is a polyhedron with 8 faces (4 triangle, triangular, 4 isosceles trapezoidal), 14 edges, and 8 vertices. As a deltahedral hexadecahedron A related construction is a hexadecahedron, 16 triangular face (ge ...

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

References

External links

The Convex Deltahedra, And the Allowance of Coplanar Faces
{{Polyhedra Polyhedra