View source for Great triambic icosahedron

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|align=center colspan=1|
|align=center colspan=1|'''Great triambic icosahedron'''
|align=center colspan=1|'''Medial triambic icosahedron'''
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|align=center colspan=1|
|align=center colspan=1|[[File:DU47 great triambic icosahedron.png|140px]]
|align=center colspan=1|[[File:DU41 medial triambic icosahedron.png|140px]]
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|bgcolor=#e7dcc3|Types
|colspan=2 align=center|Dual uniform polyhedra
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|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||colspan=2 align=center|I<sub>h</sub>
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|bgcolor=#e7dcc3|Name
|Great triambic icosahedron||Medial triambic icosahedron
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|bgcolor=#e7dcc3|[[Uniform polyhedron#Indexing|Index references]]||DU<sub>47</sub>, [[List of Wenninger polyhedron models|W]]<sub>34</sub>, 30/59||DU<sub>41</sub>, [[List of Wenninger polyhedron models|W]]<sub>34</sub>, 30/59
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|bgcolor=#e7dcc3|[[Euler characteristic|Elements]]||''F'' = 20, ''E'' = 60<BR>''V'' = 32 (χ = -8)||''F'' = 20, ''E'' = 60<BR>''V'' = 24 (χ = -16)
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|bgcolor=#e7dcc3|[[Isohedral figure|Isohedral faces]]||[[File:Great triambic icosahedron face.png|120px]]||[[File:Medial triambic icosahedron face.svg|120px]]
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|bgcolor=#e7dcc3|[[Dual polyhedron|Duals]]||[[File:Great ditrigonal icosidodecahedron.png|120px]]<BR>[[Great ditrigonal icosidodecahedron|Great ditrigonal icosidodecahedron]]||120px<BR>[[Ditrigonal dodecadodecahedron]]
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|bgcolor=#e7dcc3 colspan=3 align=center|'''[[Stellation]]'''
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|colspan=3 align=center|[[Icosahedron]]: [[List of Wenninger polyhedron models#Stellations of icosahedron|W]]<sub>34</sub>
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|colspan=3 valign=top align=center|[[File:Great triambic icosahedron stellation facets.svg|240px]]<BR>Stellation diagram
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[[File:Medial triambic icosahedron.stl|thumb|3D model of a medial triambic icosahedron]]
{{stack|[[File:Great triambic icosahedron (full).stl|thumb|3D model of a great triambic icosahedron]]}}
In [[Geometry|geometry]], the '''great triambic icosahedron''' and '''medial triambic icosahedron''' (or '''midly triambic icosahedron''') are visually identical [[Dual polyhedron|dual]] uniform polyhedra. The exterior surface also represents the '''[[The Fifty-Nine Icosahedra|De<sub>2</sub>f<sub>2</sub>]]''' [[Great triambic icosahedron#As a stellation|stellation]] of the [[Icosahedron|icosahedron]]. These figures can be differentiated by marking which intersections between edges are true [[Vertex (geometry)|vertices]] and which are not. In the above images, true vertices are marked by gold spheres, which can be seen in the concave Y-shaped areas. Alternatively, if the faces are filled with the [[Even–odd rule|even–odd rule]], the internal structure of both shapes will differ.

The 12 vertices of the [[Convex hull|convex hull]] matches the [[Vertex arrangement|vertex arrangement]] of an [[Icosahedron|icosahedron]].

== Great triambic icosahedron ==
The '''great triambic icosahedron''' is the dual of the [[Great ditrigonal icosidodecahedron|great ditrigonal icosidodecahedron]], U47. It has 20 inverted-hexagonal (triambus) faces, shaped like a three-bladed [[Engineering:Propeller|propeller]]. It has 32 vertices: 12 exterior points, and 20 hidden inside. It has 60 edges.

The faces have alternating angles of <math>\arccos(\frac{1}{4})-60^{\circ}\approx 15.522\,487\,814\,07^{\circ}</math> and <math>\arccos(-\frac{1}{4})\approx 104.477\,512\,185\,93^{\circ}</math>. The sum of the six angles is <math>360^{\circ}</math>, and not <math>720^{\circ}</math> as might be expected for a hexagon, because the polygon turns around its center twice. The [[Dihedral angle|dihedral angle]] equals <math>\arccos(-\frac{1}{3})\approx 109.471\,220\,634\,49</math>.

== Medial triambic icosahedron ==

The '''medial triambic icosahedron''' is the dual of the [[Ditrigonal dodecadodecahedron|ditrigonal dodecadodecahedron]], U41. It has 20 faces, each being simple concave [[Isotoxal figure|isotoxal]] [[Hexagon|hexagon]]s or triambi. It has 24 vertices: 12 exterior points, and 12 hidden inside. It has 60 edges.

The faces have alternating angles of <math>\arccos(\frac{1}{4})-60^{\circ}\approx 15.522\,487\,814\,07^{\circ}</math> and <math>\arccos(-\frac{1}{4})+120^{\circ}\approx 224.477\,512\,185\,93^{\circ}</math>. The [[Dihedral angle|dihedral angle]] equals <math>\arccos(-\frac{1}{3})\approx 109.471\,220\,634\,49</math>.


Unlike the great triambic icosahedron, the medial triambic icosahedron is topologically a [[Regular polyhedron|regular polyhedron]] of index two.<ref>[http://homepages.wmich.edu/~drichter/regularpolyhedra.htm The Regular Polyhedra (of index two)], David A. Richter</ref> By distorting the triambi into regular [[Hexagon|hexagon]]s, one obtains a quotient space of the [[Hyperbolic geometry|hyperbolic]] [[Order-5 hexagonal tiling|order-5 hexagonal tiling]]:

[[File:Uniform tiling 65-t0.png|250px]]

== As a stellation ==
[[File:Ninth stellation of icosahedron.png|240px]][[File:Stellation icosahedron De2f2.png|240px]]

It is Wenninger's 34th model as his [[List of Wenninger polyhedron models#Stellations of icosahedron|9th stellation of the icosahedron]]

== See also ==
* [[Triakis icosahedron]]
* Small triambic icosahedron
* Medial rhombic triacontahedron

==References==
{{reflist}}
* {{cite book | first=Magnus | last=Wenninger |  title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }}
*{{Cite book | last1=Wenninger | first1=Magnus |  title=Dual Models | publisher={{wipe|Cambridge University Press}} | isbn=978-0-521-54325-5 |mr=730208 | year=1983}}
* [[Biography:Harold Scott MacDonald Coxeter|H.S.M. Coxeter]], ''Regular Polytopes'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, 3.6 6.2 ''Stellating the Platonic solids'', pp.96-104

== External links ==
*{{mathworld|urlname= GreatTriambicIcosahedron|title= Great triambic icosahedron}}
*{{mathworld|urlname=MedialTriambicIcosahedron |title= Medial triambic icosahedron }}
* gratrix.net [http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]
* bulatov.org  [http://bulatov.org/polyhedra/dual/ud46.html Medial triambic icosahedron] [http://bulatov.org/polyhedra/dual/ud52.html Great triambic icosahedron]


[[Category:Polyhedra]]

{{Sourceattribution|Great triambic icosahedron}}