In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the triakis icosahedron is an Archimedean dual

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

 solid, or a Catalan solid

Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

. Its dual is the truncated dodecahedron

Truncated dodecahedron

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...

.

It can be seen as an icosahedron

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

 with triangular pyramids

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 augmented to each face; that is, it is the Kleetope

Kleetope

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a shallow pyramid...

 of the icosahedron. This interpretation is expressed in the name, triakis.

Related figures

This interpretation can also apply to other similar nonconvex polyhedra with pyramids of different heights:

• First stellation of icosahedron, or small triambic icosahedron
  Small triambic icosahedron
  In geometry, the small triambic icosahedron is the dual to the uniform small ditrigonal icosidodecahedron. It is composed of 20 intersecting isogonal hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8....
  
   (sometimes called a triakis icosahedron)
• Great stellated dodecahedron (with very tall pyramids)
• Great dodecahedron (with inverted pyramids)

The triakis icosahedron has numerous stellations, one of which is pictured at left.

External links

• Triakis Icosahedron -- Interactive Polyhedron Model