View source for Pentakis dodecahedron

{{short description|Catalan solid with 60 faces}}
{{Semireg dual polyhedra db|Semireg dual polyhedron stat table|dtI}}
[[File:Pentakis dodecahedron.stl|thumb|3D model of a pentakis dodecahedron]]

In [[Geometry|geometry]], a '''pentakis dodecahedron''' or '''kisdodecahedron''' is a polyhedron created by attaching a [[Pentagonal pyramid|pentagonal pyramid]] to each face of a [[Regular dodecahedron|regular dodecahedron]]; that is, it is the [[Kleetope]] of the dodecahedron. Specifically, the term typically refers to a particular [[Catalan solid]], namely the [[Dual polyhedron|dual]] of a [[Truncated icosahedron|truncated icosahedron]].

==Cartesian coordinates==

Let <math>\phi</math> be the [[Golden ratio|golden ratio]]. The 12 points given by <math>(0, \pm 1, \pm \phi)</math> and cyclic permutations of these coordinates are the vertices of a [[Regular icosahedron|regular icosahedron]]. Its dual [[Regular dodecahedron|regular dodecahedron]], whose edges intersect those of the icosahedron at right angles, has as vertices the points <math>(\pm 1, \pm 1, \pm 1)</math> together with the points <math>(\pm\phi, \pm 1/\phi, 0)</math> and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of <math>(3\phi+12)/19\approx 0.887\,057\,998\,22</math> 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 <math>2/\phi</math>. Its faces are acute isosceles triangles with one angle of <math>\arccos((-8+9\phi)/18)\approx 68.618\,720\,931\,19^{\circ}</math> and two  of <math>\arccos((5-\phi)/6)\approx 55.690\,639\,534\,41^{\circ}</math>. The length ratio between the long and short edges of these triangles equals <math>(5-\phi)/3\approx 1.127\,322\,003\,75</math>.

==Chemistry==
[[Image:C60-cpk.png|200px]]<br>The ''pentakis dodecahedron'' in a model of buckminsterfullerene: each (spherical) surface segment represents a carbon [[Physics:Atom|atom]], and if all are replaced with planar faces, a pentakis dodecahedron is produced.   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 [[Biology:Adeno-associated virus|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:
{|class=wikitable
|+ Orthogonal projections
|- align=center
!Projective<br>symmetry
|[2]
|[6]
|[10]
|-
!Image
|[[File:Dual dodecahedron t01 e66.png|120px]]
|[[File:Dual dodecahedron t01 A2.png|120px]]
|[[File:Dual dodecahedron t01 H3.png|120px]]
|-
!Dual<BR>image
|[[File:Dodecahedron t12 e66.png|120px]]
|[[File:Icosahedron t01 A2.png|120px]]
|[[File:Icosahedron t01 H3.png|120px]]
|}

== Concave pentakis dodecahedron ==
A '''concave pentakis dodecahedron''' replaces the pentagonal faces of a dodecahedron with ''inverted'' pyramids.
{| style="width: 100%;"
|- style="vertical-align: top;"
|

{{multiple image
 | align = left  | width = 200
 | image1 = Polyhedron truncated 20 dual big.png
 | image2 = Concave pentakis dodecahedron.png
 | footer = A pentakis dodecahedron (left) with inverted pyramids (right) has the same surface area.
}}
{{clear}}

== Related polyhedra==

The faces of a regular dodecahedron may be replaced (or augmented with) any regular pentagonal pyramid to produce what is in general referred to as an '''elevated dodecahedron'''. For example, if pentagonal pyramids with equilateral triangles are used, the result is a non-[[Convex polytope|convex]] [[Deltahedron|deltahedron]]. Any such elevated dodecahedron has the same combinatorial structure as a pentakis dodecahedron, i.e., the same [[Schlegel diagram]].

[[File:Spherical_pentakis_dodecahedron.png|160px|thumb|Spherical pentakis dodecahedron]]


== 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&dq=jeffrey+lindsay&pg=PA92
*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 dodecahedron.[https://www.scribd.com/doc/7817182/Doctor-Atomic-Libretto]
*In [[Software:De Blob 2|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, or at least elevated dodecahedra.

==References==
{{reflist}}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{cite web |quote=We surround the plutonium core from thirty two points spaced equally around its surface, the thirty-two points are the centers of the twenty triangular faces of an icosahedron interwoven with the twelve pentagonal faces of a dodecahedron. |url=https://www.scribd.com/doc/7817182/Doctor-Atomic-Libretto |first=Peter |last=Sellars |title=Doctor Atomic Libretto |publisher=Boosey & Hawkes |year=2005}}
*{{Cite book |last=Wenninger |first=Magnus |title=Dual Models |publisher={{wipe|Cambridge University Press}} |isbn=978-0-521-54325-5 |mr=730208 |year=1983 }} (The thirteen semiregular convex polyhedra and their duals, Page 18, Pentakisdodecahedron)
*''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Pentakis dodecahedron )

==External links==
*{{Mathworld2 |urlname=PentakisDodecahedron |title=Pentakis dodecahedron |urlname2=CatalanSolid |title2=Catalan solid}}
*[https://web.archive.org/web/20080813122248/http://polyhedra.org/poly/show/41/pentakis_dodecahedron Pentakis Dodecahedron] – Interactive Polyhedron Model
*[http://dmccooey.com/polyhedra/PentakisDodecahedron.html Visual Polyhedra pentakis dodecahedron]
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