View source for Deltoidal hexecontahedron

{{Short description|Catalan solid with 60 faces}}
{|style="float: right; border: 1px solid #BBB; margin: .5em 0 0 .5em;"
!bgcolor=#e7dcc3 colspan=2|Deltoidal hexecontahedron
|-
|align=center colspan=2|[[Image:deltoidalhexecontahedron.jpg|240px|Deltoidal hexecontahedron]]<br>[[:image:deltoidalhexecontahedron.gif|''(Click here for rotating model)'']]
|-
|bgcolor=#e7dcc3|Type||[[Catalan solid|Catalan]]
|-
|bgcolor=#e7dcc3|[[Conway polyhedron notation|Conway notation]]||oD or deD
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_f1|5|node|3|node_f1}}
|-
|bgcolor=#e7dcc3|Face polygon||[[File:DU27 facets.png|60px]]<BR>[[kite (geometry)|kite]]
|-
|bgcolor=#e7dcc3|Faces||60
|-
|bgcolor=#e7dcc3|Edges||120
|-
|bgcolor=#e7dcc3|Vertices||62 = 12 + 20 + 30
|-
|bgcolor=#e7dcc3|[[Face configuration]]||V3.4.5.4
|-
|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||[[Icosahedral symmetry|I<sub>h</sub>]], H<sub>3</sub>, [5,3], (*532)
|-
|bgcolor=#e7dcc3|[[Point groups in three dimensions#Rotation groups|Rotation group]]||I, [5,3]<sup>+</sup>, (532)
|-
|bgcolor=#e7dcc3|[[Dihedral angle]]||154.1214°<br><math>\arccos(\frac{-19-8\sqrt{5}}{41})</math>
|-
|bgcolor=#e7dcc3|Properties||convex, [[face-transitive]]
|-
|valign=top align=center|[[Image:Small rhombicosidodecahedron.png|120px]]<BR>[[rhombicosidodecahedron]]<BR>([[dual polyhedron]])
|align=center|[[Image:deltoidalhexecontahedron net.png|120px|Deltoidal hexecontahedron net]]<BR>[[Net (polyhedron)|Net]]
|}
[[File:Deltoidal hexecontahedron.stl|thumb|3D model of a deltoidal hexecontahedron]]
In [[geometry]], a '''deltoidal hexecontahedron''' (also sometimes called a ''trapezoidal hexecontahedron'', a ''strombic hexecontahedron'', or a ''tetragonal hexacontahedron''<ref>Conway, Symmetries of things, p.284-286</ref>) is a [[Catalan solid]] which is the [[dual polyhedron]] of the [[rhombicosidodecahedron]], an [[Archimedean solid]]. It is one of six Catalan solids to not have a [[Hamiltonian path]] among its vertices.<ref>{{Cite web|url=http://mathworld.wolfram.com/ArchimedeanDualGraph.html|title=Archimedean Dual Graph}}</ref>

It is topologically identical to the nonconvex [[rhombic hexecontahedron]].

==Lengths and angles==
The 60 faces are deltoids or [[kite (geometry)|kites]]. The short and long edges of each kite are in the ratio <math display="inline">1:\frac{7+\sqrt{5}}{6}</math> ≈ 1:1.539344663...

The angle between two short edges in a single face is <math display="inline">\arccos(\frac{-5-2\sqrt{5}}{20})</math> ≈ 118.2686774705°. The opposite angle, between long edges, is <math display="inline">\arccos(\frac{-5+9\sqrt{5}}{40})</math> ≈ 67.783011547435°. The other two angles of each face, between a short and a long edge each, are both equal to <math display="inline">\arccos(\frac{5-2\sqrt{5}}{10})</math> ≈ 86.97415549104°.

The dihedral angle between any pair of adjacent faces is <math display="inline">\arccos(\frac{-19-8\sqrt{5}}{41})</math> ≈ 154.12136312578°.

== Topology==
Topologically, the deltoidal hexecontahedron is identical to the nonconvex [[rhombic hexecontahedron]]. The deltoidal hexecontahedron can be derived from a [[dodecahedron]] (or [[icosahedron]]) by pushing the face centers, edge centers and vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points.

==Cartesian coordinates==
The 62 vertices of the deltoidal hexecontahedron fall in three sets centered on the origin:

* Twelve vertices are of the form of a unit [[circumradius]] [[regular icosahedron]].
* Twenty vertices are of the form of a <math>\frac{3}{11}\sqrt {15 - \frac{6}{\sqrt{5}}}\approx 0.9571</math> scaled [[regular dodecahedron]].
* Thirty vertices are of the form of a <math>3\sqrt {1-\frac{2}{\sqrt{5}}}\approx0.9748</math> scaled [[Icosidodecahedron]].

These hulls are visualized in the figure below:

:[[File:Deltoidal_Hexacontahedron_Hulls.svg|400px|Deltoidal hexacontahedron hulls]]

==Orthogonal projections==
The deltoidal hexecontahedron has 3 symmetry positions located on the 3 types of vertices:
{|class=wikitable width=640
|+ Orthogonal projections
|- align=center
!Projective<BR>symmetry
|[2]
|[2]
|[2]
|[2]
|[6]
|[10]
|-
!Image
|[[File:Dual dodecahedron t02 v.png|100px]]
|[[File:Dual dodecahedron t02 e34.png|100px]]
|[[File:Dual dodecahedron t02 e45.png|100px]]
|[[File:Dual dodecahedron t02 f4.png|100px]]
|[[File:Dual dodecahedron t02 A2.png|100px]]
|[[File:Dual dodecahedron t02 H3.png|100px]]
|-
![[rhombicosidodecahedron|Dual]]<BR>image
|[[File:Dodecahedron t02 v.png|100px]]
|[[File:Dodecahedron t02 e34.png|100px]]
|[[File:Dodecahedron t02 e45.png|100px]]
|[[File:Dodecahedron t02 f4.png|100px]]
|[[File:Dodecahedron t02 A2.png|100px]]
|[[File:Dodecahedron t02 H3.png|100px]]
|}

== Variations==
[[File:Perspectiva Corporum Regularium 41b.jpg|thumb|This figure from ''[[Perspectiva Corporum Regularium]]'' (1568) by [[Wenzel Jamnitzer]] can be seen as a deltoidal hexecontahedron.]]
The deltoidal hexecontahedron can be constructed from either the [[regular icosahedron]] or [[regular dodecahedron]] by adding vertices mid-edge, and mid-face, and creating new edges from each edge center to the face centers. [[Conway polyhedron notation]] would give these as oI, and oD, ortho-icosahedron, and ortho-dodecahedron. These geometric variations exist as a continuum along one degree of freedom.
:[[File:Deltoidal hexecontahedron on icosahedron dodecahedron.png|320px]]

== Related polyhedra and tilings ==
[[File:Spherical deltoidal hexecontahedron.png|160px|thumb|Spherical deltoidal hexecontahedron]]
{{Icosahedral truncations}}

When projected onto a sphere (see right), it can be seen that the edges make up [[Compound of dodecahedron and icosahedron|the edges of an icosahedron and dodecahedron arranged in their dual positions]].

This tiling is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.''n''.4), and continues as tilings of the [[Hyperbolic space|hyperbolic plane]]. These [[face-transitive]] figures have (*''n''32) reflectional [[Orbifold notation|symmetry]].

{{Dual expanded table}}

==See also==
*[[Deltoidal icositetrahedron]]

== References ==
{{reflist}}
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{ISBN|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal hexecontahedron)
* http://mathworld.wolfram.com/ArchimedeanDualGraph.html

==External links==
* {{Mathworld2 | urlname= DeltoidalHexecontahedron| title=DeltoidalHexecontahedron and Hamiltonian path | urlname2 =  CatalanSolid| title2 = Catalan solid}}
* [https://web.archive.org/web/20080908051203/http://polyhedra.org/poly/show/42/trapezoidal_hexecontahedron Deltoidal Hexecontahedron (Trapezoidal Hexecontrahedron)]—Interactive Polyhedron Model
* [https://web.archive.org/web/20120221170755/http://www.ict.griffith.edu.au/anthony/kites/gallery/#Ball Example in real life]—A ball almost 4 meters in diameter, from ripstop nylon, and inflated by the wind. It bounces around on the ground so that kids can play with it at kite festivals.

{{Catalan solids}}
{{Polyhedron navigator}}

[[Category:Catalan solids]]


{{Polyhedron-stub}}