geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...

. Its dual is the

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 This polyhedron can be formed from a regular dodecahedron by tr ...

Cartesian coordinates

Let

\phi

be the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. The 12 points given by

(0, \pm 1, \pm \phi)

and cyclic permutations of these coordinates are the vertices of a

regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...

. Its dual

regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...

, whose edges intersect those of the icosahedron at right angles, has as vertices the points

(\pm 1, \pm 1, \pm 1)

together with the points

(\pm\phi, \pm 1/\phi, 0)

and cyclic permutations of these coordinates. Multiplying all coordinates of this dodecahedron by a factor of

(7\phi-1)/11\approx 0.938\,748\,901\,93

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

2

. Its faces are isosceles triangles with one obtuse angle of

\arccos(-3\phi/10)\approx 119.009\,350\,869\,29^

and two acute ones of

\arccos((\phi+7)/10)\approx 30.480\,324\,565\,36^

. The length ratio between the long and short edges of these triangles equals

(\phi+7)/5\approx 1.723\,606\,797\,75

Orthogonal projections

The triakis icosahedron has three symmetry positions, two on vertices, and one on a midedge: The Triakis icosahedron has five special

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

s, 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 plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

Kleetope

It can be seen as an

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

triangular pyramid In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

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

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. Kleetopes are named after Victor Klee. Exam ...

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

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: * ''First stellation of icosahedron'', or

Small triambic icosahedron In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8. It is an isohedron, meaning that all of its faces ar ...

, or sometimes called a ''Triakis icosahedron'' (among others) *

Great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each ve ...

(with very tall pyramids) *

Great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...

(with inverted pyramids)

Stellations

The triakis icosahedron has numerous

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...

s, 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 In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

figures have (*n32) reflectional

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

References

* (Section 3-9) * * (The thirteen semiregular convex polyhedra and their duals, Page 19, Triakisicosahedron) *''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,

(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis icosahedron )

External links

*
Triakis Icosahedron
– Interactive Polyhedron Model Catalan solids {{Polyhedron-stub