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Pentakis dodecahedron

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Pentakis dodecahedron

Pentakis dodecahedron

(Click here for rotating model)
Type Catalan solid
Coxeter diagram
Conway notation kD
Face type V5.6.6

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
(dual polyhedron)
Pentakis dodecahedron Net
Net

In geometry, a pentakis dodecahedron or kisdodecahedron 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. [1] There are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include:

  • As the heights of the pentagonal pyramids are raised, at a certain point adjoining pairs of triangular faces merge to become rhombi, and the shape becomes a rhombic triacontahedron.
  • As the height is raised further, the shape becomes non-convex. In particular, an equilateral or deltahedron version of the pentakis dodecahedron, which has sixty equilateral triangular faces as shown in the adjoining figure, is slightly non-convex due to its taller pyramids (note, for example, the negative dihedral angle at the upper left of the figure).
A non-convex variant with equilateral triangular faces.

Other more non-convex geometric variants include:

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

Contents

  • Chemistry 1
  • Biology 2
  • Orthogonal projections 3
  • Related polyhedra 4
  • Cultural references 5
  • References 6
  • External links 7

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:

Orthogonal projections
Projective
symmetry
[2] [6] [10]
Image
Dual
image

Related polyhedra

Spherical pentakis dodecahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
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
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact hyperb. Parac. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Truncated
figures
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6

Cultural references

  • Spaceship Earth in Walt Disney World's Epcot is based on one.
  • The model for a campus arts workshop designed by Jeffrey Lindsay was actually a hemispherical pentakis dodecahedron http://books.google.co.uk/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 dodecahedron.[3]
  • 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

  1. ^ Conway, Symmetries of things, p.284
  • (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, ISBN 978-1-56881-220-5 [4] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Pentakis dodecahedron )

External links

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