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 ....

, a pentakis dodecahedron is 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 icosahedron

Truncated icosahedron

In geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges....

, an Archimedean solid

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...

.

It can be seen as a dodecahedron with a pentagonal pyramid

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 . Like any pyramid, it is self-dual....

 covering 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 dodecahedron. This interpretation is expressed in its name.

Its construction is topologically the same as that of:

1. The small stellated dodecahedron (with very tall pyramids).
2. Great pentakis dodecahedron
   Great pentakis dodecahedron
   In geometry, the great pentakis dodecahedron is a nonconvex isohedral polyhedron.It is the dual of the uniform small stellated truncated dodecahedron. The decagrammic faces pass close to the origin in the uniform polyhedron, causing this dual to be very spikey....
   
    (with extremely tall pyramids)
3. Wenninger's third stellation of icosahedron (with inverted pyramids).

If one affixes pentagrammic pyramids into Wenninger's third stellation of icosahedron one obtains the great icosahedron.

The pentakis dodecahedron in a model of buckminsterfullerene

Buckminsterfullerene

Buckminsterfullerene is a spherical fullerene molecule with the formula . It was first intentionally prepared in 1985 by Harold Kroto, James Heath, Sean O'Brien, Robert Curl and Richard Smalley at Rice University...

: each surface segment represents a carbon

Carbon

Carbon is the chemical element with symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalent—making four electrons available to form covalent chemical bonds...

 atom

Atom

The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

.

• Spaceship Earth
  Spaceship Earth (Disney)
  Spaceship Earth is the iconic and symbolic structure of Epcot, a theme park that is part of the Walt Disney World Resort. One of the most recognizable structures at the Walt Disney World Resort, it is not only the centerpiece and main focal point of Epcot, but also the name of the attraction housed...
  
   in Walt Disney World's Epcot
  Epcot
  Epcot is a theme park in the Walt Disney World Resort, located near Orlando, Florida. The park is dedicated to the celebration of human achievement, namely international culture and technological innovation. The second park built at the resort, it opened on October 1, 1982 and was initially named...
  
   is based on one.
• The shape of the 'Crystal Dome' used in the popular TV game show The Crystal Maze
  The Crystal Maze
  The Crystal Maze was a British game show, produced by Chatsworth Television and shown on Channel 4 in the United Kingdom between 15 February 1990 and 10 August 1995. There was one series per year, with the first four series presented by Richard O'Brien and the final two by Ed Tudor-Pole, who made...
  
  was based on a pentakis dodecahedron.
• In Doctor Atomic
  Doctor Atomic
  Doctor Atomic is an opera by the contemporary American composer John Adams, with libretto by Peter Sellars. It premiered at the San Francisco Opera on 1 October 2005. The work focuses on the great stress and anxiety experienced by those at Los Alamos while the test of the first atomic bomb was...
  
  , the shape of the first atomic bomb detonated in New Mexico
  New Mexico
  New Mexico is a state located in the southwest and western regions of the United States. New Mexico is also usually considered one of the Mountain States. With a population density of 16 per square mile, New Mexico is the sixth-most sparsely inhabited U.S...
  
   was a pentakis dodecahedron.http://www.scribd.com/doc/7817182/Doctor-Atomic-Libretto
• In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron.