# 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}}}
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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} ) [/itex]|

|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=|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"
$\arccos ( -\frac{4}{5} )$| |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=|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"
$\arccos ( -\frac{24 + 15\sqrt{5}}{61} )$| |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=|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"
$\arccos ( -\frac{80 + 9\sqrt{5}}{109} )$| |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=
|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=|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=|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.  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).
160px

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:

Projectivesymmetry Image Dualimage    120px 120px 120px 120px 120px 120px

## Related polyhedra

File:Spherical pentakis dodecahedron.png
Spherical pentakis dodecahedron
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
Uniform dual figures
n-kis
figures
Config.
50px
V2.6.6
50px
V3.6.6
50px
V4.6.6
50px
V5.6.6
50px
V6.6.6
50px
V7.6.6
50px
V8.6.6
50px
V∞.6.6
V12i.6.6 V9i.6.6 V6i.6.6 V3i.6.6
Coxeter