Pentakis dodecahedron

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Pentakis dodecahedron
Pentakis dodecahedron
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 5.pngCDel node.png
Conway notation kD
Face type V5.6.6
DU25 facets.png

isosceles triangle
Faces 60
Edges 90
Vertices 32
Vertices by type 20{6}+12{5}
Symmetry group Ih, H3, [5,3], (*532)
Rotation group I, [5,3]+, (532)
Dihedral angle 156° 43' 7"
 \arccos ( -\frac{80 + 9\sqrt{5}}{109} )
Properties convex, face-transitive
Truncated icosahedron.png
Truncated icosahedron
(dual polyhedron)
Pentakis dodecahedron Net
Net

In geometry, a pentakis dodecahedron is a Catalan solid. Its dual is the truncated icosahedron, an Archimedean solid.

It can be seen as a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope of the dodecahedron. This interpretation is expressed in its name.

Geometric variations[edit]

Its construction is topologically the same as that of:

  1. Small stellated dodecahedron.png The small stellated dodecahedron (with very tall pyramids).
  2. DU58 great pentakisdodecahedron.png Great pentakis dodecahedron (with extremely tall pyramids)
  3. Third stellation of icosahedron.png 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.

Chemistry[edit]

C60-cpk.png
The pentakis dodecahedron in a model of buckminsterfullerene: each surface segment represents a carbon atom.

Related polyhedra[edit]

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg POV-Ray-Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5
Dimensional family of truncated polyhedra and tilings: n.6.6
Symmetry
*n42
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
Order 12 24 48 120
Truncated
figures
Hexagonal dihedron.png
2.6.6
Uniform tiling 332-t12.png
3.6.6
Uniform tiling 432-t12.png
4.6.6
Uniform tiling 532-t12.png
5.6.6
Uniform tiling 63-t12.png
6.6.6
Uniform tiling 73-t12.png
7.6.6
Uniform tiling 83-t12.png
8.6.6
H2 tiling 23i-6.png
∞.6.6
Coxeter
Schläfli
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node.png
t{3,2}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t{3,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t{3,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t{3,5}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
t{3,6}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 7.pngCDel node.png
t{3,7}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 8.pngCDel node.png
t{3,8}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel infin.pngCDel node.png
t{3,∞}
Uniform dual figures
n-kis
figures
Hexagonal Hosohedron.svg
V2.6.6
Triakistetrahedron.jpg
V3.6.6
Tetrakishexahedron.jpg
V4.6.6
Pentakisdodecahedron.jpg
V5.6.6
Uniform tiling 63-t2.png
V6.6.6
Order3 heptakis heptagonal til.png
V7.6.6
Uniform dual tiling 433-t012.png
V8.6.6
H2checkers 33i.png
V∞.6.6
Coxeter CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel infin.pngCDel node.png

Cultural references[edit]

  • Spaceship Earth in Walt Disney World's Epcot is based on one.
  • 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.[1]
  • 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[edit]

  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Sellars, Peter (2005). "Doctor Atomic Libretto". Boosey & Hawkes. "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." 
  • Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 978-0-521-54325-5. MR 730208.  (The thirteen semiregular convex polyhedra and their duals, Page 18, Pentakisdodecahedron)
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [2] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Pentakis dodecahedron )

External links[edit]