View source for Truncated great icosahedron

{{Short description|Polyhedron with 32 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|gtI}}
[[File:Truncated great icosahedron.stl|thumb|3D model of a truncated great icosahedron]]

In [[Geometry|geometry]], the '''truncated great icosahedron''' (or '''great truncated icosahedron''') is a nonconvex uniform polyhedron, indexed as U<sub>55</sub>. It has 32 faces (12 [[Pentagram|pentagram]]s and 20 [[Hexagon|hexagon]]s), 90 edges, and 60 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/55.html|title=55: great truncated icosahedron|last=Maeder|first=Roman|website=MathConsult}}</ref> It is given a [[Schläfli symbol]] {{math|t{3,{{Frac|5|2}}} }} or {{math|t<sub>0,1</sub>{3,{{Frac|5|2}}} }} as a [[Truncation (geometry)|truncated]] [[Great icosahedron|great icosahedron]].

== Cartesian coordinates ==
Cartesian coordinates for the vertices of a ''truncated great icosahedron'' centered at the origin are all the [[Parity of a permutation|even permutations]] of

<math display=block>\begin{array}{crccc}
  \Bigl(& \pm\,1,& 0,& \pm\,\frac{3}{\varphi} &\Bigr) \\
  \Bigl(& \pm\,2,& \pm\,\frac{1}{\varphi},& \pm\,\frac{1}{\varphi^3} &\Bigr) \\
  \Bigl(& \pm \bigl[1+\frac{1}{\varphi^2}\bigr],& \pm\,1,& \pm\,\frac{2}{\varphi} &\Bigr)
\end{array}</math>

where <math>\varphi = \tfrac{1+\sqrt 5}{2}</math> is the [[Golden ratio|golden ratio]]. Using <math>\tfrac{1}{\varphi^2} = 1 - \tfrac{1}{\varphi}</math> one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to <math>10-\tfrac{9}{\varphi}.</math> The edges have length 2.

== Related polyhedra ==
This polyhedron is the [[Truncation (geometry)|truncation]] of the [[Great icosahedron|great icosahedron]]:

The [[Truncation (geometry)|truncated]] ''great stellated dodecahedron'' is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a [[Great dodecahedron|great dodecahedron]] inscribed within and sharing the edges of the icosahedron.

{| class="wikitable" width=500
!Name
![[Great stellated dodecahedron|Great<BR>stellated<BR>dodecahedron]]
!Truncated great stellated dodecahedron
![[Great icosidodecahedron|Great<BR>icosidodecahedron]]
!Truncated<BR>great<BR>icosahedron
![[Great icosahedron|Great<BR>icosahedron]]
|- align=center
!Coxeter-Dynkin<BR>diagram
|{{CDD|node|3|node|5|rat|d2|node_1}}
|{{CDD|node|3|node_1|5|rat|d2|node_1}}
|{{CDD|node|3|node_1|5|rat|d2|node}}
|{{CDD|node_1|3|node_1|5|rat|d2|node}}
|{{CDD|node_1|3|node|5|rat|d2|node}}
|-
!Picture
|[[Image:Great stellated dodecahedron.png|100px]]
|[[Image:Icosahedron.png|100px]]
|[[Image:Great icosidodecahedron.png|100px]]
|[[Image:Great truncated icosahedron.png|100px]]
|[[Image:Great icosahedron.png|100px]]
|}

{{-}}

=== Great stellapentakis dodecahedron ===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|gtI}}
[[File:Great stellapentakis dodecahedron.stl|thumb|3D model of a great stellapentakis dodecahedron]]
The '''great stellapentakis dodecahedron''' is a nonconvex [[Isohedral figure|isohedral]] [[Polyhedron|polyhedron]]. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.

== See also ==
* [[List of uniform polyhedra]]

==References==
{{Reflist}}
*{{Citation | last1=Wenninger | first1=Magnus |  title=Dual Models | publisher={{wipe|Cambridge University Press}} | isbn=978-0-521-54325-5 |mr=730208 | year=1983 | doi=10.1017/CBO9780511569371}}

== External links ==
* {{mathworld | urlname = GreatTruncatedIcosahedron| title = Truncated great icosahedron}}
* {{mathworld | urlname = GreatStellapentakisDodecahedron| title =Great stellapentakis dodecahedron}}
* [http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]
[[File:Great_stellated_dodecahedron_truncations.gif|right|thumb|240x240px|Animated truncation sequence from {{{frac|5|2}}, 3} to {3, {{frac|5|2}}}]]
{{Nonconvex polyhedron navigator}}

[[Category:Uniform polyhedra]]



{{Sourceattribution|Truncated great icosahedron}}