Editing Truncated icosidodecahedron

{{short description|Archimedean solid}}
{{Semireg polyhedra db|Semireg polyhedron stat table|grID}}

In [[geometry]], a '''truncated icosidodecahedron''', '''rhombitruncated icosidodecahedron''',<ref name="wenninger16">Wenninger Model Number 16</ref> '''great rhombicosidodecahedron''',<ref name="williams94">Williams (Section 3-9, p. 94)</ref><ref name="cromwell82">Cromwell (p. 82)</ref> '''omnitruncated dodecahedron''' or '''omnitruncated icosahedron'''<ref name="johnson1966">Norman Woodason Johnson, "The Theory of Uniform Polytopes and Honeycombs", 1966</ref> is an [[Archimedean solid]], one of thirteen [[Convex polytope|convex]], [[Isogonal figure|isogonal]], non-[[Prism (geometry)|prismatic]] solids constructed by two or more types of [[regular polygon|regular]] polygon [[Face (geometry)|face]]s.

It has 62 faces: 30 [[square (geometry)|squares]], 20 regular [[hexagon]]s, and 12 regular [[decagon]]s. It has the most edges and vertices of all [[Platonic solid|Platonic]] and Archimedean solids, though the [[snub dodecahedron]] has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a [[Circumscribed sphere|sphere]] in which it is [[inscribed]], very narrowly beating the snub dodecahedron (89.63%) and small [[rhombicosidodecahedron]] (89.23%), and less narrowly beating the [[truncated icosahedron]] (86.74%); it also has by far the greatest volume (206.8 cubic units) when its edge length equals 1. Of all [[vertex-transitive]] polyhedra that are not prisms or [[antiprism]]s, it has the largest sum of angles (90 + 120 + 144 = 354 degrees) at each vertex; only a prism or antiprism with more than 60 sides would have a larger sum. Since each of its faces has [[point symmetry]] (equivalently, 180° [[rotational symmetry]]), the truncated icosidodecahedron is a '''15'''-[[zonohedron]].

==Names==
{|
| <!-- Does someone know a better way to make the image float right, but not below the infobox? -->
The name ''truncated icosidodecahedron'', given originally by [[Johannes Kepler]], is misleading. An actual [[truncation (geometry)|truncation]] of an [[icosidodecahedron]] has [[rectangle]]s instead of [[square]]s. This nonuniform polyhedron is [[topologically]] equivalent to the Archimedean solid.

Alternate interchangeable names are:
*''Truncated icosidodecahedron'' ([[Johannes Kepler]])
*''Rhombitruncated icosidodecahedron'' ([[Magnus Wenninger]]<ref name="wenninger16" />)
*''Great rhombicosidodecahedron'' ([[Robert Williams (geometer)|Robert Williams]],<ref name="williams94" /> Peter Cromwell<ref name="cromwell82" />)
*''[[Omnitruncation (geometry)|Omnitruncated]] dodecahedron'' or ''icosahedron'' ([[Norman Johnson (mathematician)|Norman Johnson]]<ref name="johnson1966" />)
|style="padding-left:50px;"| <!-- right table cell -->
{{multiple image
 | align = left  | width = 150
 | image1 = Polyhedron 12-20 big.png
 | image2 = Polyhedron nonuniform truncated 12-20 big.png
 | footer = Icosidodecahedron and its truncation
}}
|}

The name ''great rhombicosidodecahedron'' refers to the relationship with the (small) [[rhombicosidodecahedron]] (compare section [[#Dissection|Dissection]]).<br>
There is a [[nonconvex uniform polyhedron]] with a similar name, the [[nonconvex great rhombicosidodecahedron]].

==Area and volume==
The surface area ''A'' and the volume ''V'' of the truncated icosidodecahedron of edge length ''a'' are:{{citation needed|date=January 2017}}
<!-- wrong value, given on MathWorld, need a source for correct value above
<math>\begin{align} A & = 30 \left [ 1 + \sqrt{ 2 \left ( 4 + \sqrt{5} + \sqrt{15+6\sqrt{6}} \right ) } \right ] a^2 \\
& \approx 175.031\,045a^2 \\-->
: <math>\begin{align} A &= 30 \left (1 + \sqrt{3} + \sqrt{5 + 2\sqrt{5}} \right)a^2 &&\approx 174.292\,0303a^2. \\ V &= \left( 95 + 50\sqrt{5} \right) a^3 &&\approx 206.803\,399a^3. \end{align}</math>

If a set of all 13 [[Archimedean solid]]s were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.

==Cartesian coordinates==
[[Cartesian coordinates]] for the vertices of a truncated icosidodecahedron with edge length 2''φ''&nbsp;−&nbsp;2, centered at the origin, are all the [[even permutation]]s of:<ref>{{mathworld|title=Icosahedral group|urlname=IcosahedralGroup}}</ref>
:(±{{sfrac|1|''φ''}}, ±{{sfrac|1|''φ''}}, ±(3&nbsp;+&nbsp;''φ'')),
:(±{{sfrac|2|''φ''}}, ±''φ'', ±(1&nbsp;+&nbsp;2''φ'')),
:(±{{sfrac|1|''φ''}}, ±''φ''<sup>2</sup>, ±(−1&nbsp;+&nbsp;3''φ'')),
:(±(2''φ''&nbsp;−&nbsp;1), ±2, ±(2&nbsp;+&nbsp;''φ'')) and
:(±''φ'', ±3, ±2''φ''),
where ''φ''&nbsp;=&nbsp;{{sfrac|1 + {{sqrt|5}}|2}} is the [[golden ratio]].

== Dissection ==

The truncated icosidodecahedron is the [[convex hull]] of a [[rhombicosidodecahedron]] with [[cuboid]]s above its 30 squares, whose height to base ratio is {{math|[[golden ratio|φ]]}}. The rest of its space can be dissected into nonuniform cupolas, namely 12 [[pentagonal cupola|between inner pentagons and outer decagons]] and 20 [[triangular cupola|between inner triangles and outer hexagons]].

An alternative dissection also has a rhombicosidodecahedral core. It has 12 [[pentagonal rotunda]]e between inner pentagons and outer decagons. The remaining part is a [[toroidal polyhedron]].

{| class="wikitable collapsible"
! dissection images
|-
|
{{multiple image
 | align = left  | total_width = 650
 | image1 = Small in great rhombi 12-20, davinci small with cuboids.png |width1=1|height1=1 
 | image2 = Small in great rhombi 12-20, davinci.png
 | footer = These images show the rhombicosidodecahedron (violet) and the truncated icosidodecahedron (green). If their edge lengths are 1, the distance between corresponding squares is {{math|[[golden ratio|φ]]}}.
}}
[[File:Toroidal_excavated_truncated_Icosidodecahedron.gif|320px|thumb|right|The toroidal polyhedron remaining after the core and twelve rotundae are cut out]]
|}

==Orthogonal projections==
The truncated icosidodecahedron has seven special [[orthogonal projection]]s, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. 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>4-6
!Edge<br>4-10
!Edge<br>6-10
!Face<br>square
!Face<br>hexagon
!Face<br>decagon
|-
!Solid
|
|
|
|
|[[File:Polyhedron great rhombi 12-20 from blue max.png|100px]]
|[[File:Polyhedron great rhombi 12-20 from yellow max.png|100px]]
|[[File:Polyhedron great rhombi 12-20 from red max.png|100px]]
|-
!Wireframe
|[[File:Dodecahedron_t012_v.png|100px]]
|[[File:Dodecahedron_t012_e46.png|100px]]
|[[File:Dodecahedron_t012_e4x.png|100px]]
|[[File:Dodecahedron_t012_e6x.png|100px]]
|[[File:Dodecahedron_t012_f4.png|100px]]
|[[File:Dodecahedron_t012_A2.png|100px]]
|[[File:Dodecahedron_t012_H3.png|100px]]
|- align=center
!Projective<br>symmetry
|[2]<sup>+</sup>
|[2]
|[2]
|[2]
|[2]
|[6]
|[10]
|-
!Dual<BR>image
|[[File:Dual dodecahedron_t012_v.png|100px]]
|[[File:Dual dodecahedron_t012_e46.png|100px]]
|[[File:Dual dodecahedron_t012_e4x.png|100px]]
|[[File:Dual dodecahedron_t012_e6x.png|100px]]
|[[File:Dual dodecahedron_t012_f4.png|100px]]
|[[File:Dual dodecahedron_t012_A2.png|100px]]
|[[File:Dual dodecahedron_t012_H3.png|100px]]
|}

==Spherical tilings and Schlegel diagrams ==
The truncated icosidodecahedron 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=3|[[Stereographic projection]]s
|-
!
![[Decagon]]-centered
![[Hexagon]]-centered
![[Square]]-centered
|- align=center valign=top
|[[Image:Uniform tiling 532-t012.png|160px]]
|[[Image:Truncated icosidodecahedron stereographic projection decagon.png|160px]]
|[[Image:Truncated icosidodecahedron stereographic projection hexagon.png|168px]]
|[[Image:Truncated icosidodecahedron stereographic projection square.png|172px]]
|}

== Geometric variations ==
Within [[Icosahedral symmetry]] there are unlimited geometric variations of the ''truncated icosidodecahedron'' with [[Isogonal figure|isogonal]] faces. The [[truncated dodecahedron]], [[rhombicosidodecahedron]], and [[truncated icosahedron]] as degenerate limiting cases.
{| class=wikitable
|[[File:Truncated dodecahedron.png|100px]]
|[[File:Great truncated icosidodecahedron convex hull.png|100px]]
|[[File:Nonuniform_truncated_icosidodecahedron.png|100px]]
|[[File:Uniform_polyhedron-53-t012.png|100px]]
|[[File:Truncated_dodecadodecahedron_convex_hull.png|100px]]
|[[File:Icositruncated_dodecadodecahedron_convex_hull.png|100px]]
|[[File:Truncated icosahedron.png|100px]]
|[[File:Small rhombicosidodecahedron.png|100px]]
|- align=center
|{{CDD|node_1|5|node_1|3|node}}
|colspan=5|{{CDD|node_1|5|node_1|3|node_1}}
|{{CDD|node|5|node_1|3|node_1}}
|{{CDD|node_1|5|node|3|node_1}}
|}

== Truncated icosidodecahedral graph ==
{{Infobox graph
 | name = Truncated icosidodecahedral graph
 | image = [[File:Truncated icosidodecahedral graph.png|240px]]
 | image_caption = 5-fold symmetry
 | namesake = 
 | vertices = 120 
 | edges = 180 
 | automorphisms = 120 (A<sub>5</sub>×2)
 | radius = 15
 | diameter = 15
 | girth = 4
 | 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 icosidodecahedral graph''' (or '''great rhombicosidodecahedral graph''') is the [[1-skeleton|graph of vertices and edges]] of the truncated icosidodecahedron, one of the [[Archimedean solid]]s. It has 120 [[Vertex (graph theory)|vertices]] and 180 edges, and is a [[zero-symmetric graph|zero-symmetric]] and [[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
|+ [[Schlegel diagram]] graphs
|- align=center
|[[File:Truncated icosidodecahedral graph-hexcenter.png|160px]]<BR>3-fold symmetry
|[[File:Truncated icosidodecahedral graph-squarecenter.png|200px]]<BR>2-fold symmetry
|}
{{Clear}}

== Related polyhedra and tilings==
{| class=wikitable align=right width=320
|[[File:Conway_polyhedron_b3I.png|160px]] 
|[[File:Conway_polyhedron_b3D.png|160px]]
|-
|colspan=2|Bowtie icosahedron and dodecahedron contain two trapezoidal faces in place of the square.<ref>[http://www.cgl.uwaterloo.ca/csk/papers/bridges2001.html Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons] Craig S. Kaplan</ref>
|}
{{Icosahedral truncations}}

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2''p'') and [[Coxeter-Dynkin diagram]] {{CDD|node_1|p|node_1|3|node_1}}.  For ''p''&nbsp;<&nbsp;6, the members of the sequence are [[Omnitruncation (geometry)|omnitruncated]] polyhedra ([[zonohedron]]s), shown below as spherical tilings. For ''p''&nbsp;>&nbsp;6, they are tilings of the hyperbolic plane, starting with the [[truncated triheptagonal tiling]].

{{Omnitruncated table}}

==Notes==
{{reflist}}

==References==
*{{Citation |last1=Wenninger |first1=Magnus |author1-link=Magnus Wenninger |title=Polyhedron Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-09859-5 |mr=0467493 |year=1974}}
*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79–86 ''Archimedean solids''|isbn=0-521-55432-2}}
*{{The Geometrical Foundation of Natural Structure (book)}}
*Cromwell, P.; [https://books.google.com/books?id=OJowej1QWpoC&pg=PA82 ''Polyhedra''], CUP hbk (1997), pbk. (1999).
*{{mathworld2 |urlname=GreatRhombicosidodecahedron |title=GreatRhombicosidodecahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x5x - grid}}

==External links==
* {{mathworld | urlname = GreatRhombicosidodecahedron | title = Great rhombicosidodecahedron}}
* * {{mathworld | urlname = GreatRhombicosidodecahedralGraph | title = Great rhombicosidodecahedral graph}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=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&name=Truncated+Icosidodecahedron#applet Editable printable net of a truncated icosidodecahedron 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

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{{DEFAULTSORT:Truncated Icosidodecahedron}}
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