View source for Truncated dodecahedron

{{Semireg polyhedra db|Semireg polyhedron stat table|tD}}
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.

==Geometric relations==
This [[polyhedron]] can be formed from a [[dodecahedron]] by [[Truncation (geometry)|truncating]] (cutting off) the corners so the [[pentagon]] faces become [[decagon]]s and the corners become [[triangle]]s.

It is used in the [[cell-transitive]] hyperbolic space-filling tessellation, the [[bitruncated icosahedral honeycomb]].

==Area and volume==
The area ''A'' and the [[volume]] ''V'' of a truncated dodecahedron of edge length ''a'' are:
:<math>\begin{align}
A &= 5 \left(\sqrt{3}+6\sqrt{5+2\sqrt{5}}\right) a^2 &&\approx 100.990\,76a^2 \\
V &= \tfrac{5}{12} \left(99+47\sqrt{5}\right) a^3 &&\approx 85.039\,6646a^3
\end{align}</math>

==Cartesian coordinates==
[[Cartesian coordinates]] for the vertices of a [[Truncation (geometry)|truncated]] [[dodecahedron]] with edge length 2''φ''&nbsp;−&nbsp;2, centered at the origin,<ref>{{mathworld |title=Icosahedral group |urlname=IcosahedralGroup}}</ref> are all even permutations of:
:(0, ±{{sfrac|1|''φ''}}, ±(2&nbsp;+&nbsp;''φ''))
:(±{{sfrac|1|''φ''}}, ±''φ'', ±2''φ'')
:(±''φ'', ±2, ±(''φ''&nbsp;+&nbsp;1))

where ''φ''&nbsp;=&nbsp;{{sfrac|1 + {{sqrt|5}}|2}} is the [[golden ratio]].

==Orthogonal projections==
The ''truncated dodecahedron'' has five special [[orthogonal projection]]s, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A<sub>2</sub> and H<sub>2</sub> [[Coxeter plane]]s.
{|class=wikitable
|+ Orthogonal projections
|-
!Centered by
!Vertex
!Edge<br>3-10
!Edge<br>10-10
!Face<br>Triangle
!Face<br>Decagon
|-
!Solid
|
|
|[[File:Polyhedron truncated 12 from blue max.png|120px]]
|[[File:Polyhedron truncated 12 from yellow max.png|120px]]
|[[File:Polyhedron truncated 12 from red max.png|120px]]
|-
!Wireframe
|[[File:Dodecahedron_t01_v.png|120px]]
|[[File:Dodecahedron_t01_e3x.png|120px]]
|[[File:Dodecahedron_t01_exx.png|120px]]
|[[File:Dodecahedron_t01_A2.png|120px]]
|[[File:Dodecahedron_t01_H3.png|120px]]
|- align=center
!Projective<br>symmetry
|[2]
|[2]
|[2]
|[6]
|[10]
|-
!Dual
|[[File:Dual dodecahedron_t12_v.png|120px]]
|[[File:Dual dodecahedron_t12_e3x.png|120px]]
|[[File:Dual dodecahedron_t12_exx.png|120px]]
|[[File:Dual dodecahedron_t12_A2.png|120px]]
|[[File:Dual dodecahedron_t12_H3.png|120px]]
|}

==Spherical tilings and Schlegel diagrams==
The truncated dodecahedron can also be represented as a [[spherical tiling]], and projected onto the plane via a [[stereographic projection]]. This projection is [[Conformal map|conformal]], preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

[[Schlegel diagram]]s are similar, with a [[perspective projection]] and straight edges.
{|class=wikitable
|-
![[Orthographic projection]]
!colspan=2|[[Stereographic projection]]s
|- align=center valign=top
|[[Image:Uniform tiling 532-t01.png|160px]]
|[[Image:truncated dodecahedron stereographic projection decagon.png|160px]]<br>[[Decagon]]-centered
|[[Image:truncated dodecahedron stereographic projection triangle.png|160px]]<br>[[Triangle]]-centered
|-
|[[File:Truncated_dodecahedron_ortho-color.png|160px]]
|[[File:Truncated_dodecahedron_schlegel.png|160px]]
|[[File:Truncated_dodecahedron_schlegel-tricenter.png|160px]]
|}

== Vertex arrangement==
It shares its [[vertex arrangement]] with three [[nonconvex uniform polyhedra]]:
{|class="wikitable" width="400" style="vertical-align:top;text-align:center"
|[[Image:Truncated dodecahedron.png|100px]]<br>Truncated dodecahedron
|[[Image:Great icosicosidodecahedron.png|100px]]<br>[[Great icosicosidodecahedron]]
|[[Image:Great ditrigonal dodecicosidodecahedron.png|100px]]<br>[[Great ditrigonal dodecicosidodecahedron]]
|[[Image:Great dodecicosahedron.png|100px]]<br>[[Great dodecicosahedron]]
|}

== Related polyhedra and tilings ==

It is part of a truncation process between a dodecahedron and icosahedron:
{{Icosahedral truncations}}

This polyhedron is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[vertex configuration]]s (3.2''n''.2''n''), and [''n'',3] [[Coxeter group]] symmetry.

{{Truncated figure1 small table}}

== Truncated dodecahedral graph ==
{{Infobox graph
 | name = Truncated dodecahedral graph
 | image = [[File:Truncated dodecahedral graph.png|240px]]
 | image_caption = 5-fold symmetry [[schlegel diagram]]
 | namesake = 
 | vertices = 60
 | edges = 90
 | automorphisms = 120
 | radius = 
 | diameter = 
 | girth = 
 | chromatic_number = 2
 | chromatic_index = 
 | fractional_chromatic_index = 
 | properties = [[Cubic graph|Cubic]], [[hamiltonian graph|Hamiltonian]], [[regular graph|regular]], [[Zero-symmetric graph|zero-symmetric]]
}}
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]].<ref>{{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=[[Oxford University Press]]|year= 1998|page=269}}</ref>

{| class=wikitable
|- align=center
|[[File:Truncated Dodecahedral Graph.svg|240px]]<BR>Circular
|}
{{-}}

==See also==
*[[:Image:Truncatedhexahedron.gif|Spinning truncated cube]]
*[[Cube-connected cycles]], a family of graphs that includes the [[skeleton (topology)|skeleton]] of the truncated cube

==Notes==
{{reflist}}

==References==
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{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]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra

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

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