Editing Truncated dodecahedron

{{Short description|Archimedean solid with 32 faces}}
{{CS1 config|mode=cs1}}
{{infobox polyhedron
 | name = Truncated dodecahedron
 | image = Dwunastościan ścięty.svg
 | type = [[Archimedean solid]]
 | symmetry = [[icosahedral symmetry]] <math> \mathrm{I}_\mathrm{h} </math>
 | dual = [[Triakis icosahedron]]
 | faces = 32 | edges = 90 | vertices = 60
 | angle = 10-10: 116.57° <br> 3-10: 142.62°
 | vertex_figure = Polyhedron truncated 12 vertfig.svg
 | net = Polyhedron truncated 12 net.svg
}}
In [[geometry]], the '''truncated dodecahedron''' is an [[Archimedean solid]]. It has 12 regular [[decagon]]al faces, 20 regular [[triangular]] faces, 60 vertices and 90 edges.

== Construction ==
The truncated dodecahedron is constructed from a [[regular dodecahedron]] by cutting all of its vertices off, a process known as [[Truncation (geometry)|truncation]].{{r|ziya}} Alternatively, the truncated dodecahedron can be constructed by [[expansion (geometry)|expansion]]: pushing away the edges of a regular dodecahedron, forming the [[Pentagon|pentagonal]] faces into [[Decagon|decagonal]] faces, as well as the vertices into [[triangle]]s.{{r|vxac}} Therefore, it has 32 faces, 90 edges, and 60 vertices.{{r|berman}}

The truncated dodecahedron may also be constructed by using [[Cartesian coordinate]]s. With an edge length <math> 2\varphi - 2 </math> centered at the origin, they are all even permutations of
<math display="block">
\left(0, \pm \frac{1}{\varphi}, \pm (2 + \varphi) \right), \qquad
\left(\pm \frac{1}{\varphi}, \pm \varphi, \pm 2 \varphi \right), \qquad
\left(\pm \varphi, \pm 2, \pm (\varphi + 1) \right),
</math>
where <math display="inline"> \varphi = \frac{1 + \sqrt{5}}{2} </math> is the [[golden ratio]].<ref>{{mathworld |title=Icosahedral group |urlname=IcosahedralGroup}}</ref>

== Properties ==
The surface area <math> A </math> and the volume <math> V </math> of a truncated dodecahedron of edge length <math> a </math> are:{{r|berman}}
<math display="block"> \begin{align}
A &= 5 \left(\sqrt{3}+6\sqrt{5+2\sqrt{5}}\right) a^2 &&\approx 100.991a^2 \\
V &= \frac{5}{12} \left(99+47\sqrt{5}\right) a^3 &&\approx 85.040a^3
\end{align}</math>

The dihedral angle of a truncated dodecahedron between two regular dodecahedral faces is 116.57°, and that between triangle-to-dodecahedron is 142.62°.{{r|johnson}}

[[File:Truncated dodecahedron.stl|thumb|3D model of a truncated dodecahedron]]
The truncated dodecahedron is an [[Archimedean solid]], meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.<ref name=diudea/> It has the same symmetry as the regular icosahedron, the [[icosahedral symmetry]].<ref name=kk/> The polygonal faces that meet for every vertex are one equilateral triangle and two regular decagon, and the [[vertex figure]] of a truncated dodecahedron is <math> 3 \cdot 10^2 </math>. The dual of a truncated dodecahedron is [[triakis icosahedron]], a [[Catalan solid]],{{r|williams}} which shares the same symmetry as the truncated dodecahedron.{{r|holden}}

The truncated dodecahedron is non-[[Chirality (mathematics)|chiral]], meaning it is congruent to its mirror image.{{r|kk}}

== Truncated dodecahedral graph ==
[[File:Truncated dodecahedral graph.png|thumb|240px|The graph of a truncated dodecahedron]]
In the [[mathematics|mathematical]] field of [[graph theory]], a ''truncated dodecahedral graph'' is the [[1-skeleton|graph of vertices and edges]] of the ''truncated dodecahedron'', one of the [[Archimedean solid]]s. It has 60 [[Vertex (graph theory)|vertices]] and 90 edges, and is a [[cubic graph|cubic]] [[Archimedean graph]].{{r|rw}}

== Related polyhedron ==
The truncated dodecahedron can be applied in the polyhedron's construction known as the [[Augmentation (geometry)|augmentation]]. Examples of polyhedrons are the [[Johnson solid]]s, whose constructions are involved by attaching [[pentagonal cupola]]s onto the truncated dodecahedron: [[augmented truncated dodecahedron]], [[parabiaugmented truncated dodecahedron]], [[metabiaugmented truncated dodecahedron]], and [[triaugmented truncated dodecahedron]].{{r|berman}}

== See also ==
* [[Great stellated truncated dodecahedron]]

==References==
{{reflist|refs=

<ref name=berman>{{cite journal
 | last = Berman | first = Martin
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971| issue = 5
}} See in particular page 336.</ref>

<ref name=diudea>{{cite book
 | last = Diudea | first = M. V.
 | year = 2018
 | title = Multi-shell Polyhedral Clusters
 | series = Carbon Materials: Chemistry and Physics
 | volume = 10
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 978-3-319-64123-2
 | doi = 10.1007/978-3-319-64123-2
 | url = https://books.google.com/books?id=p_06DwAAQBAJ
 | page = [https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]
}}</ref>

<ref name=holden>{{cite book
 | last = Holden | first = Alan
 | title = Shapes, Space, and Symmetry
 | series = Dover Books on Mathematics
 | publisher = [[Courier Corporation]]
 | year = 1991
 | isbn = 9780486268514
 | url = https://books.google.com/books?id=p_06DwAAQBAJ
 | page = [https://books.google.com/books?id=VFiTF-fXI20C&pg=PA52 52]
}}</ref>

<ref name=johnson>{{cite journal
 | last = Johnson | first = Norman W. | author-link = Norman Johnson (mathematician)
 | doi = 10.4153/cjm-1966-021-8
 | journal = [[Canadian Journal of Mathematics]]
 | mr = 0185507
 | pages = 169–200
 | title = Convex polyhedra with regular faces
 | volume = 18
 | year = 1966
 | zbl = 0132.14603
}}</ref>

<ref name=kk>{{cite conference
 | last1 = Koca | first1 = M.
 | last2 = Koca | first2 = N. O.
 | year = 2013
 | title = Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27&ndash;31 October 2010
 | contribution = Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes
 | contribution-url = https://books.google.com/books?id=ILnBkuSxXGEC
 | publisher = World Scientific
 |page=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 48]
}}</ref>

<ref name=rw>{{cite book
 | last1 = Read | first1 = R. C.
 | last2 = Wilson | first2 = R. J.
 | title = An Atlas of Graphs
 | publisher = [[Oxford University Press]]
 | year = 1998
 | page = 269
}}</ref>

<ref name=vxac>{{cite book
 | last1 = Viana | first1 = Vera
 | last2 = Xavier | first2 = João Pedro
 | last3 = Aires | first3 = Ana Paula
 | last4 = Campos | first4 = Helena
 | year = 2019
 | editor-last = Cocchiarella | editor-first = Luigi
 | title = ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics: 40th Anniversary - Milan, Italy, August 3-7, 2018
 | contribution = Interactive Expansion of Achiral Polyhedra
 | contribution-url = https://books.google.com/books?id=rEpjDwAAQBAJ&pg=PA1122
 | page = 1122
 | doi = 10.1007/978-3-319-95588-9
 | isbn = 978-3-319-95588-9
}}</ref>

<ref name=williams>{{cite book
 | last = Williams | first = Robert | authorlink = Robert Williams (geometer)
 | year = 1979
 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design
 | publisher = Dover Publications, Inc.
 | url = https://archive.org/details/geometricalfound00will
 | page = [https://archive.org/details/geometricalfound00will/page/88/mode/1up?view=theater 88]
| isbn = 978-0-486-23729-9 }}</ref>

<ref name=ziya>{{cite journal
 | last = Ziya | first = Ümit
 | year = 2019
 | title = Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces 
 | journal = Cumhuriyet Science Journal
 | volume = 40 | issue = 2 | pages = 457&ndash;470
 | doi = 10.17776/csj.534616 
}}</ref>

}}

==Further reading==
*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79–86 ''Archimedean solids''|isbn=0-521-55432-2}}

==External links==
*{{mathworld2 | urlname = TruncatedDodecahedron| title = Truncated dodecahedron | urlname2 =  ArchimedeanSolid  | title2 =  Archimedean solid}}
**{{mathworld | urlname = TruncatedDodecahedralGraph| title = Truncated dodecahedral graph}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3x5x - tid}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1e9V3YL5nW2MMkIcdn0TdMHHhXMiuoCQGqz2g3IjH7orIJ5iBy9LQ80CKQP1GAP9MmtklgzVBcF5ZfK9LsPLcjTfCVtbQWJrpIJTarRzJGitPNEnHrk3rNm5pr6Gzui1P5MD7RwSrFu6TKzjy5qQl5PYokM9mcFWcoPivzjQxlRGa1eVpVmZl5Uv2nXTaX5RSgc2N5B3daPbsAUEsCGxrnbgMLCKvMvztIjl44GGTstwl3pC589OwhVUTHvkTzg6b4dpshGHQn4ajtxQA8chKkqzW1wKBsKuMpbqE4oCXbIi2sfEgppN1tcDBWVOJUXQfPiEglU1jtQi7fUj5xDW2PpZtdwQDmwpC3Lk&name=Truncated+Dodecahedron#applet Editable printable net of a truncated dodecahedron with interactive 3D view]

{{Archimedean solids}}
{{Polyhedron navigator}}

[[Category:Uniform polyhedra]]
[[Category:Archimedean solids]]
[[Category:Truncated tilings]]