Triakis icosahedron


In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.

Cartesian coordinates

Let be the golden ratio. The 12 points given by and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points together with the points and cyclic permutations of these coordinates. Multiplying all coordinates of this dodecahedron by a factor of gives a slightly smaller dodecahedron. The 20 vertices of this dodecahedron, together with the vertices of the icosahedron, are the vertices of a triakis icosahedron centered at the origin. The length of its long edges equals. Its faces are isosceles triangles with one obtuse angle of and two acute ones of. The length ratio between the long and short edges of these triangles equals.

Orthogonal projections

The triakis icosahedron has three symmetry positions, two on vertices, and one on a midedge:
The Triakis icosahedron
has five special orthogonal projections, centered on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A2 and H2 Coxeter planes.
Projective
symmetry
Image
Dual
image

Kleetope

It can be seen as an icosahedron with triangular pyramids augmented to each face; that is, it is the Kleetope of the icosahedron. This interpretation is expressed in the name, triakis.
If the icosahedron is augmented by tetrahedral without removing the center icosahedron, one gets the net of an icosahedral pyramid.

Other triakis icosahedra

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

The triakis icosahedron has numerous stellations, including this one.

Related polyhedra

The triakis icosahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry.