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Pentakis dodecahedron

{{{dtI-name}}}
[[image:{{{dtI-image2}}}|280px]]
[[:image:{{{dtI-image3}}}|(Click here for rotating model)]]
Type Catalan solid
Coxeter diagram {{{dtI-Cox}}}
Conway notation {{{dtI-conway}}}
Face type {{{dtI-ffig}}}[[image:{{{dtI-fimage}}}|right|60px]]
{{{dtI-ftype}}}
Faces {{{dtI-F}}}
Edges {{{dtI-E}}}
Vertices {{{dtI-V}}}
Vertices by type {{{dtI-Vdetail}}}
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Rotation group {{{dtI-rotgroup}}}
Dihedral angle {{{dtI-dihedral}}}
Properties convex, face-transitive {{{dtI-special}}}
[[Image:{{{dtI-dimage}}}|100px]]
[[{{{dtI-dual}}}]]
(dual polyhedron)
[[image:{{{dtI-netimage}}}|100px|Pentakis dodecahedron Net]]
Net
{17} ) </math>|

|dtC-special=

|dtO-name=Tetrakis hexahedron|dtO-image=tetrakishexahedron.jpg|dtO-image2=tetrakishexahedron.jpg|dtO-image3=tetrakishexahedron.gif|dtO-dimage=Truncated octahedron.png|dtO-netimage=tetrakishexahedron_net.png| |dtO-Cox=File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 4.pngFile:CDel node.png|dtO-conway=kC| |dtO-V=14|dtO-E=36|dtO-F=24|dtO-Vdetail=6{4}+8{6}|dtO-chi=2| |dtO-ffig=V4.6.6|dtO-ftype=isosceles triangle|dtO-fimage=DU08 facets.png |dtO-group=Oh, B3, [4,3], (*432)| |dtO-rotgroup=O, [4,3]+, (432)| |dtO-dual=Truncated octahedron|dtO-dihedral=143° 7' 48"
<math> \arccos ( -\frac{4}{5} ) </math>| |dtO-special=

|dtD-name=Triakis icosahedron|dtD-image=triakisicosahedron.jpg|dtD-image2=triakisicosahedron.jpg|dtD-image3=triakisicosahedron.gif|dtD-dimage=Truncated dodecahedron.png|dtD-netimage=triakisicosahedron_net.png| |dtD-Cox=File:CDel node f1.pngFile:CDel 5.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png|dtD-conway=kI| |dtD-V=32|dtD-E=90|dtD-F=60|dtD-Vdetail=20{3}+12{10}|dtD-chi=2| |dtD-ffig=V3.10.10|dtD-ftype=isosceles triangle|dtD-fimage=DU26 facets.png |dtD-group=Ih, H3, [5,3], (*532)| |dtD-rotgroup=I, [5,3]+, (532)| |dtD-dual=Truncated dodecahedron|dtD-dihedral=160° 36' 45"
<math> \arccos ( -\frac{24 + 15\sqrt{5}}{61} ) </math>| |dtD-special=

|dtI-name=Pentakis dodecahedron|dtI-image=pentakisdodecahedron.jpg|dtI-image2=pentakisdodecahedron.jpg|dtI-image3=pentakisdodecahedron.gif|dtI-dimage=Truncated icosahedron.png|dtI-netimage=pentakisdodecahedron_net.png| |dtI-Cox=File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 5.pngFile:CDel node.png|dtI-conway=kD| |dtI-V=32|dtI-E=90|dtI-F=60|dtI-Vdetail=20{6}+12{5}|dtI-chi=2| |dtI-ffig=V5.6.6|dtI-ftype=isosceles triangle|dtI-fimage=DU25 facets.png |dtI-group=Ih, H3, [5,3], (*532)| |dtI-rotgroup=I, [5,3]+, (532)| |dtI-dual=Truncated icosahedron|dtI-dihedral=156° 43' 7"
<math> \arccos ( -\frac{80 + 9\sqrt{5}}{109} ) </math>| |dtI-special=

|dCO-name=Rhombic dodecahedron|dCO-image=rhombicdodecahedron.jpg|dCO-image2=rhombicdodecahedron.jpg|dCO-image3=rhombicdodecahedron.gif|dCO-dimage=cuboctahedron.png|dCO-netimage=rhombicdodecahedron_net.svg| |dCO-Cox=File:CDel node f1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node f1.png
File:CDel node.pngFile:CDel 4.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png|dCO-conway=jC| |dCO-V=14|dCO-E=24|dCO-F=12|dCO-Vdetail=8{3}+6{4}|dCO-chi=2| |dCO-ffig=V3.4.3.4|dCO-ftype=rhombus|dCO-fimage=DU07 facets.png |dCO-group=Oh, B3, [4,3], (*432)| |dCO-rotgroup=O, [4,3]+, (432)| |dCO-dual=Cuboctahedron|dCO-dihedral=120°| |dCO-special=edge-transitive, parallelohedron

|dID-name=Rhombic triacontahedron|dID-image=rhombictriacontahedron.png|dID-image2=rhombictriacontahedron.svg|dID-image3=rhombictriacontahedron.gif|dID-dimage=icosidodecahedron.svg|dID-netimage=rhombictriacontahedron net.svg| |dID-Cox=File:CDel node.pngFile:CDel 5.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png|dID-conway=jD| |dID-V=32|dID-E=60|dID-F=30|dID-Vdetail=20{3}+12{5}|dID-chi=2| |dID-ffig=V3.5.3.5|dID-ftype=rhombus|dID-fimage=DU24 facets.png |dID-group=Ih, H3, [5,3], (*532)| |dID-rotgroup=I, [5,3]+, (532)| |dID-dual=Icosidodecahedron|dID-dihedral=144°| |dID-special=edge-transitive, zonohedron

|dSD-name=Pentagonal hexecontahedron|dSD-image=Pentagonalhexecontahedron.jpg|dSD-image2=Pentagonalhexecontahedron.jpg|dSD-image3=Pentagonalhexecontahedronccw.gif|dSD-dimage=Snub_dodecahedron_ccw.png|dSD-netimage=Pentagonalhexecontahedron net.png| |dSD-Cox=File:CDel node fh.pngFile:CDel 5.pngFile:CDel node fh.pngFile:CDel 3.pngFile:CDel node fh.png|dSD-conway=gD| |dSD-V=92|dSD-E=150|dSD-F=60|dSD-Vdetail=12 {5}
20+60 {3}|dSD-chi=2| |dSD-ffig=V3.3.3.3.5|dSD-ftype=irregular pentagon|dSD-fimage=DU29 facets.png |dSD-group=I, ½H3, [5,3]+, (532)| |dSD-rotgroup=I, [5,3]+, (532)| |dSD-dual=Snub dodecahedron|dSD-dihedral=153° 10' 43"| |dSD-special=chiral

}} 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

  • An equilateral or deltahedron version of the pentakis dodecahedron has sixty equilateral triangular faces. This version, shown in the red figure below, is slightly non-convex due to its taller pyramids (note, for example, the negative dihedral angle at the upper left of the figure).
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Other more non-convex geometric variants include:

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

Chemistry

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

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 120px 120px 120px
Dual
image
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Related polyhedra

File:Spherical pentakis dodecahedron.png
Spherical pentakis dodecahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node h.pngFile:CDel 5.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png
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{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
File:CDel node f1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 5.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 5.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 5.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngFile:CDel 5.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node fh.pngFile:CDel 5.pngFile:CDel node fh.pngFile:CDel 3.pngFile:CDel node fh.png
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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 spherical polyhedra and tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact hyperb. Parac. Noncompact hyperbolic
*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]
 
[12i,3] [9i,3] [6i,3] [3i,3]
Figures 50px 50px 50px 50px 50px 50px 50px 50px 50px 50px 50px 50px
Schläfli t{3,2} t{3,3} t{3,4} t{3,5} t{3,6} t{3,7} t{3,8} t{3,∞} t{3,12i} t{3,9i} t{3,6i} t{3,3i}
Coxeter File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 7.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 8.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel ultra.pngFile:CDel node.png
Uniform dual figures
n-kis
figures
Config.
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V2.6.6
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V3.6.6
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V4.6.6
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V5.6.6
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V6.6.6
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V7.6.6
50px
V8.6.6
50px
V∞.6.6
V12i.6.6 V9i.6.6 V6i.6.6 V3i.6.6
Coxeter File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 2.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 4.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 5.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 6.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 7.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 8.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel infin.pngFile:CDel node.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel ultra.pngFile:CDel node.png

Cultural references

References

  1. ^ Conway, Symmetries of things, p.284
  • 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

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