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 dual of an Archimedean solid, in this case, the truncated icosahedron.

Cartesian coordinates

Let $\phi$ be the golden ratio. The 12 points given by $(0, \pm 1, \pm \phi)$ 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 $(\pm 1, \pm 1, \pm 1)$ together with the points $(\pm\phi, \pm 1/\phi, 0)$ and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of $(3\phi+12)/19\approx 0.887\,057\,998\,22$ gives a slightly smaller icosahedron. The 12 vertices of this icosahedron, together with the vertices of the dodecahedron, are the vertices of a pentakis dodecahedron centered at the origin. The length of its long edges equals $2/\phi$. Its faces are acute isosceles triangles with one angle of $\arccos((-8+9\phi)/18)\approx 68.618\,720\,931\,19^$ and two of $\arccos((5-\phi)/6)\approx 55.690\,639\,534\,41^$. The length ratio between the long and short edges of these triangles equals $(5-\phi)/3\approx 1.127\,322\,003\,75$.

Chemistry

The ''pentakis dodecahedron'' in a model of buckminsterfullerene: each surface segment represents a carbon atom. Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom.

Biology

The ''pentakis dodecahedron'' is also a model of some icosahedrally symmetric viruses, such as Adeno-associated virus. These have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a ''pentakis dodecahedron''.

Orthogonal projections

The pentakis dodecahedron has three symmetry positions, two on vertices, and one on a midedge:

Concave pentakis dodecahedron

A concave pentakis dodecahedron adds inverted pyramids on the pentagonal faces of a dodecahedron.
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Related polyhedra

See also

* Excavated dodecahedron

Cultural references

*The Spaceship Earth structure at Walt Disney World's Epcot is a derivative of a pentakis dodecahedron.
*The model for a campus arts workshop designed by Jeffrey Lindsay was actually a hemispherical pentakis dodecahedron https://books.google.com/books?id=JD8EAAAAMBAJ&pg=PA92&dq=jeffrey+lindsay&hl=en&ei=oF88Tv25F7OisQLGwbwt&sa=X&oi=book_result&ct=result&redir_esc=y#v=onepage&q=jeffrey%20lindsay&f=false
*The shape of the "Crystal Dome" used in the popular TV game show ''The Crystal Maze'' was based on a pentakis dodecahedron.
*In Doctor Atomic, the shape of the first atomic bomb detonated in New Mexico was a pentakis dodecahedro

*In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron. These Domes also appear whenever the player transforms on a dome in the Hypno Ray level.
*Some Geodomes in which people play on are Pentakis Dodecahedra.

References

* (Section 3-9)
*
* (The thirteen semiregular convex polyhedra and their duals, Page 18, Pentakisdodecahedron)
*''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, Pentakis dodecahedron )

External links

*
Pentakis Dodecahedron
– Interactive Polyhedron Model

{{Polyhedron navigator

Catalan solids
Geodesic polyhedra