View source for Truncated great dodecahedron

{{Short description|Polyhedron with 24 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|tgD}}
[[File:Truncated great dodecahedron.stl|thumb|3D model of a truncated great dodecahedron]]
In [[Geometry|geometry]], the '''truncated great dodecahedron''' is a nonconvex uniform polyhedron, indexed as U<sub>37</sub>. It has 24 faces (12 [[Pentagram|pentagrams]] and 12 [[Decagon|decagons]]), 90 edges, and 60 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/37.html|title=37: truncated great dodecahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}}</ref> It is given a [[Schläfli symbol]] t{5,5/2}.

== Related polyhedra ==

It shares its [[Vertex arrangement|vertex arrangement]] with three other [[Uniform polyhedron|uniform polyhedra]]: the [[Nonconvex great rhombicosidodecahedron|nonconvex great rhombicosidodecahedron]], the [[Great dodecicosidodecahedron|great dodecicosidodecahedron]], and the [[Great rhombidodecahedron|great rhombidodecahedron]]; and with the uniform compounds of [[Compound of six pentagonal prisms|6]] or [[Compound of twelve pentagonal prisms|12 pentagonal prisms]].

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[Image:Uniform great rhombicosidodecahedron.png|100px]]<BR>[[Nonconvex great rhombicosidodecahedron]]
| [[Image:Great dodecicosidodecahedron.png|100px]]<BR>[[Great dodecicosidodecahedron]]
| [[Image:Great rhombidodecahedron.png|100px]]<BR>[[Great rhombidodecahedron]]
|-
| [[Image:Great truncated dodecahedron.png|100px]]<BR>'''Truncated great dodecahedron'''
| [[Image:UC34-6 pentagonal prisms.png|100px]]<BR>[[Compound of six pentagonal prisms]]
| [[Image:UC35-12 pentagonal prisms.png|100px]]<BR>[[Compound of twelve pentagonal prisms]]
|}
This polyhedron is the [[Truncation (geometry)|truncation]] of the [[Great dodecahedron|great dodecahedron]]:

The [[Truncation (geometry)|truncated]] small stellated dodecahedron looks like a [[Dodecahedron|dodecahedron]] on the surface, but it has 24 faces, 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams).

{| class="wikitable" width=500
!Name
![[Small stellated dodecahedron]]
![[Dodecahedron|Truncated small stellated dodecahedron]]
![[Dodecadodecahedron]]
!Truncated<BR>great<BR>dodecahedron
![[Great dodecahedron|Great<BR>dodecahedron]]
|- align=center
!Coxeter-Dynkin<BR>diagram
|{{CDD|node|5|node|5|rat|d2|node_1}}
|{{CDD|node|5|node_1|5|rat|d2|node_1}}
|{{CDD|node|5|node_1|5|rat|d2|node}}
|{{CDD|node_1|5|node_1|5|rat|d2|node}}
|{{CDD|node_1|5|node|5|rat|d2|node}}
|- align=center
!Picture
|[[Image:Small stellated dodecahedron.png|100px]]
|[[Image:Dodecahedron.png|100px]]
|[[Image:Dodecadodecahedron.png|100px]]
|[[Image:Great truncated dodecahedron.png|100px]]
|[[Image:Great dodecahedron.png|100px]]
|}
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=== Small stellapentakis dodecahedron ===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|tgD}}
[[File:Small stellapentakis dodecahedron.stl|thumb|3D model of a small stellapentakis dodecahedron]]
The [[Small stellapentakis dodecahedron|small stellapentakis dodecahedron]] (or '''small astropentakis dodecahedron''') is a nonconvex [[Isohedral figure|isohedral]] [[Polyhedron|polyhedron]]. It is the [[Dual polyhedron|dual]] of the truncated great dodecahedron. 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 = TruncatedGreatDodecahedron | title = Truncated great dodecahedron}}
* {{mathworld | urlname = SmallStellapentakisDodecahedron| title =Small stellapentakis dodecahedron}}
* [http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]
[[File:Small stellated dodecahedron truncations.gif|thumb|right|240px|Animated truncation sequence from {{{frac|5|2}}, 5} to {5, {{frac|5|2}}}]]
{{Nonconvex polyhedron navigator}}

[[Category:Nonconvex polyhedra]]
[[Category:Uniform polyhedra]]



{{Sourceattribution|Truncated great dodecahedron}}