In
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 pentakis icosidodecahedron or
subdivided icosahedron is a
convex polyhedron
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
with 80 triangular
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
, 120
edges, and 42
vertices. It is a
dual of the ''truncated rhombic triacontahedron'' (
chamfered dodecahedron).
Construction
Its name comes from a topological construction from the
icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...
with the
kis operator applied to the pentagonal faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general icosahedral symmetry can be maintained even with the 12 order-5 vertices at a different distance from the center as the other 30.
It can also be topologically constructed from the
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 ...
, dividing each triangular face into 4 triangles by adding mid-edge vertices. From this construction, all 80 triangles will be equilateral, but faces will be
coplanar
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...
.
Related polyhedra
File:Icosidodecahedron.png, Icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...
File:Pentakisdodecahedron.jpg, Pentakis dodecahedron
In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that it ...
is a slightly smaller 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 so ...
which has 60 isosceles triangle faces, 90 edges (2 types), and 32 vertices (2 types).
File:StellaTripentakisIcosidodecahedron.png, Tripentakis icosidodecahedron, 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 icosidodecahedron, can be obtained by raising low pyramids on each equilateral triangular face on a pentakis icosidodecahedron. It has 120 isosceles triangle faces (2 types), 180 edges (3 types) and 62 vertices (3 types).
File:Small icosihemidodecahedron.png, The nonconvex small icosihemidodecahedron looks like a pentakis icosidodecahedron with inverted pentagonal pyramid
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual.
The ''regular'' pentagonal pyramid has a base that is a r ...
s meeting at the polyhedron center.
Related polytopes
It represents the exterior envelope of a vertex-centered
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 ...
of the
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...
, one of six
convex regular 4-polytopes, into 3 dimensions.
See also
*
Tetrakis cuboctahedron
In geometry, the tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It is a dual of the truncated rhombic dodecahedron.
Its name comes from a topological construction from the cuboctahedron wi ...
References
*
George W. Hart, ''Sculpture based on Propellorized Polyhedra'', Proceedings of MOSAIC 2000, Seattle, WA, August, 2000, pp. 61–7
*
John Horton Conway, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008,
** Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings (p 284)
* Dover 1999 {{ISBN, 978-0-486-40921-4
External links
VTML polyhedral generatorTry "k5aD" (
Conway polyhedron notation)
Geodesic polyhedra