View source for Pentagonal hexecontahedron

{{Short description|Catalan solid with 60 faces}}
{{infobox polyhedron
 | name = Pentagonal hexacontahedron
 | image = Pentagonalhexecontahedron.jpg
 | faces = 60
 | edges = 150
 | vertices	= 92
 | symmetry = [[icosahedral symmetry]]
 | dual = [[snub dodecahedron]]
 | angle = 153.2&deg;
 | net = Pentagonalhexecontahedron net.svg
}}

In [[geometry]], a '''pentagonal hexecontahedron''' is a [[Catalan solid]], dual of the [[snub dodecahedron]]. It has two distinct forms, which are [[mirror image]]s (or "[[Chirality (mathematics)|enantiomorph]]s") of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices. Among the Catalan and [[Archimedean solid|Archimedean]] solids, it has the second largest number of vertices, after the [[truncated icosidodecahedron]], which has 120 vertices.

== Properties ==
[[File:Pentagonal hexecontahedron.stl|thumb|3D model of a pentagonal hexecontahedron]]
The faces are irregular [[pentagon]]s with two long edges and three short edges. Let <math>\xi\approx 0.943\,151\,259\,24</math> be the real zero of the polynomial <math>x^3+2x^2-\phi^2</math>. Then the ratio <math>l</math> of the edge lengths is given by:<math display="block"> l = \frac{1+\xi}{2-\xi^2}\approx 1.749\,852\,566\,74.</math>The faces have four equal obtuse angles and one acute angle (between the two long edges). The obtuse angles equal <math>\arccos(-\xi/2)\approx 118.136\,622\,758\,62^{\circ}</math>, and the acute one equals <math>\arccos(-\phi^2\xi/2+\phi)\approx 67.453\,508\,965\,51^{\circ}</math>. The dihedral angle equals <math>\arccos(-\xi/(2-\xi))\approx 153.2^{\circ}</math>.

Note that the face centers of the [[snub dodecahedron]] cannot serve directly as vertices of the pentagonal hexecontahedron: the four triangle centers lie in one plane but the pentagon center does not; it needs to be radially pushed out to make it coplanar with the triangle centers. Consequently, the vertices of the pentagonal hexecontahedron do not all lie on the same sphere and by definition it is not a [[zonohedron]].

To find the volume and surface area of a pentagonal hexecontahedron, denote the shorter side of one of the pentagonal faces as <math>b</math>, and set a constant<ref>{{Cite web|title=Pentagonal Hexecontahedron - Geometry Calculator|url=https://rechneronline.de/pi/pentagonal-hexecontahedron.php|website=rechneronline.de|access-date=2020-05-26}}</ref><math display="block"> t = \frac{\sqrt[3]{44+12\phi(9+\sqrt{81\phi-15})}+\sqrt[3]{44+12\phi(9-\sqrt{81\phi-15})}-4}{12} \approx 0.472. </math>Then the surface area (<math> A </math>) is:<math display="block"> A = \frac{30b^2\cdot(2+3t)\cdot\sqrt{1-t^2}}{1-2t^2}\approx162.698b^2.</math>And the volume (<math> V </math>) is:<math display="block"> V = \frac{5b^3(1+t)(2+3t)}{(1-2t^2)\cdot\sqrt{1-2t}}\approx189.789b^3. </math>Using these, one can calculate the measure of [[sphericity]] for this shape:<math display="block"> \Psi = \frac{\pi^{\frac{1}{3}}(6V)^{\frac{2}{3}}}{A} \approx 0.982 </math>

== Construction ==
[[File:Pentagonal_Hexacontahedron_Convex_Hulls.svg|thumb|299px|Combining a unit [[circumradius]] [[icosahedron]] (12) centered at the origin with a chiral [[snub dodecahedron]] (60) combined with a [[dodecahedron]] of the same non-unity circumradius (20) to construct the pentagonal hexecontahedron]]

The pentagonal hexecontahedron can be constructed from a snub dodecahedron without taking the dual. Pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal hexecontahedron.<ref>[https://groups.google.com/g/sci.math/c/c9e37XmKO24/m/okJW1M54kc4J Reference]</ref>

An alternate construction method uses [[quaternions]] and the [[icosahedral symmetry]] of the [[Weyl group]] orbits <math>O(\Lambda)=W(H_3)/C_2 \approx A_5=I</math> of order 60.<ref>{{cite arXiv
|title=Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions
|first1=Mehmet
|last1=Koca
|first2=Nazife
|last2=Ozdes Koca
|first3=Muna
|last3=Al-Shu’eilic
|year=2011
|class=math-ph
|eprint=1006.3149
}}</ref> This is shown in the figure on the right.

Specifically, with quaternions from the binary Icosahedral group <math>(p,q) \in I_h</math>, where <math>q=\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then just as the Coxeter group  <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the [[600-cell]] and the [[120-cell]] of order 14400, we have <math>W(H_3)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace=A_5\times C_2=I_h</math> of order 120. <math>I </math> is defined as the even permutations of <math>I_h</math> such that <math>[I,\bar I]:r </math> gives the 60 twisted chiral snub dodecahedron coordinates, where <math>r\approx -0.389662 e_1 + 0.267979 e_2 -0.881108 e_3 </math> is one permutation from the first set of 12 in those listed [[Pentagonal_hexecontahedron#Cartesian_coordinates|above]]. The exact coordinate for  <math>r</math> is obtained by taking the solution to <math>x^3-x^2-x-\phi=0</math>, with <math>x\approx 1.94315</math>, and applying it to the normalization of <math>r=(-1+x^2(-1-2/\phi-x\phi)e_1  + (3-x^2+3x\phi)e_2 + ((x^3-1/\phi)\phi^3)e_3</math>.

=== Cartesian coordinates ===
Using the [[Icosahedral symmetry]] in the orbits of the Weyl group <math>O(\Lambda)=W(H_3)/C_2 \approx A_5</math>of order 60<ref>{{cite journal
|title=Catalan Solids Derived From 3D-Root Systems and Quaternions
|first1=Mehmet
|last1=Koca
|first2=Nazife
|last2=Ozdes Koca
|first3=Ramazon
|last3=Koc
|year=2010
|journal=Journal of Mathematical Physics
|volume=51
|issue=4
|doi=10.1063/1.3356985 
|arxiv=0908.3272
|s2cid=115157829
}}</ref>  gives the following Cartesian coordinates with <math>\phi=\frac{1+\sqrt{5}}{2}</math> is the [[golden ratio]]:

* Twelve vertices of a [[regular icosahedron]] with unit circumradius centered at the origin with the coordinates <math display=block>\frac{(0, \pm 1, \pm \phi)}{\sqrt{\phi^2 + 1}}       ,        \frac{(\pm 1, \pm \phi, 0)}{\sqrt{\phi^2 + 1}}       ,        \frac{(\pm \phi, 0, \pm 1)}{\sqrt{\phi^2 + 1}}.</math>
* Twenty vertices of [[regular dodecahedron]] of unit circumradius centered at the origin scaled by a factor<math>R\approx 0.95369785218</math> from the exact solution to the equation <math>700569 - 1795770 x^2 + 1502955 x^4 - 423900 x^6 + 14175 x^8 - 2250 x^{10} + 125 x^{12} = 0</math>, which gives the coordinates 
<math display=block>(\pm 1, \pm 1,\pm 1)\frac{R}{\sqrt{3}} </math> and <math display=block>(0, \pm \phi, \pm \frac{1}{\phi})\frac{R}{\sqrt{3}} , (\pm \frac{1}{\phi}, 0 , \pm \phi) \frac{R}{\sqrt{3}} ,  (\pm \phi, \pm \frac{1}{\phi},0)\frac{R}{\sqrt{3}}.</math>
* Sixty vertices of a  unit circumradius [[chiral]] [[snub dodecahedron]] scaled by <math>R</math>. There are five sets of twelve vertices, all with even [[permutation]]s (i.e. with a [[parity of a permutation|parity signature]]=1). 
A group of two sets of twelve have 0 or 2 minus signs (i.e. 1 or 3 plus signs): 
<math display=block>(\pm 0.267979, \pm 0.881108, \pm 0.389662) R ,</math>
<math display=block>(\pm 0.721510, \pm 0.600810, \pm 0.344167) R ,</math>
and another group of three sets of 12 have 0 or 2 plus signs (i.e. 1 or 3 minus signs):<math display=block>(\pm 0.176956, \pm 0.824852, \pm 0.536941) R ,</math> 
<math display=block>(\pm 0.435190, \pm0.777765,  \pm 0.453531) R ,</math>
<math display=block>(\pm 0.990472, \pm 0.103342, \pm 0.091023) R .</math>
Negating all vertices in both groups gives the mirror of the chiral snub dodecahedron, yet results in the same pentagonal hexecontahedron convex hull.

== Variations==
[[Isohedral figure|Isohedral]] variations can be constructed with pentagonal faces with 3 edge lengths.

This variation shown can be constructed by adding pyramids to 12 pentagonal faces and 20 triangular faces of a [[snub dodecahedron]] such that the new triangular faces are coparallel to other triangles and can be merged into the pentagon faces.
{| class=wikitable width=480
|- align=center valign=top
|[[File:Pentagonal_hexecontahedron_variation0.png|160px]]<BR>[[Snub dodecahedron]] with augmented pyramids and merged faces
|[[File:Pentagonal_hexecontahedron_variation.png|160px]]<BR>Example variation
|[[File:Pentagonal_hexecontahedron_variation_net.png|160px]]<BR>Net
|}

==Orthogonal projections==
The ''pentagonal hexecontahedron'' has three symmetry positions, two on vertices, and one mid-edge.
{|class=wikitable width=500
|+ Orthogonal projections
|- align=center
!Projective<BR>symmetry
|[3]
|[5]<sup>+</sup>
|[2]
|-
!Image
|[[Image:Dual snub dodecahedron A2.png|120px]]
|[[Image:Dual snub dodecahedron H2.png|120px]]
|[[Image:Dual snub dodecahedron e1.png|120px]]
|-
!Dual<BR>image
|[[Image:Snub dodecahedron A2.png|120px]]
|[[Image:Snub dodecahedron H2.png|120px]]
|[[Image:Snub dodecahedron e1.png|120px]]
|}

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

This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with [[face configuration]]s (V3.3.3.3.''n''). (The sequence progresses into tilings the hyperbolic plane to any ''n''.) These [[face-transitive]] figures have (n32) rotational [[Orbifold notation|symmetry]].
{{Snub table}}

== See also==
*[[Truncated pentagonal hexecontahedron]]
*[[Amazon Spheres]]

==References==
{{Reflist}}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{Citation |last=Wenninger |first=Magnus |authorlink=Magnus Wenninger |title=Dual Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-54325-5 |mr=730208 |year=1983 |doi=10.1017/CBO9780511569371}} (The thirteen semiregular convex polyhedra and their duals, Page 29, Pentagonal hexecontahedron)
*''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 287, pentagonal hexecontahedron )

==External links==
*{{Mathworld2 |urlname=PentagonalHexecontahedron |title=Pentagonal hexecontahedron |urlname2=CatalanSolid |title2=Catalan solid}}
*[https://web.archive.org/web/20080924091620/http://polyhedra.org/poly/show/44/pentagonal_hexecontahedron Pentagonal Hexecontrahedron] – Interactive Polyhedron Model

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

[[Category:Catalan solids]]
[[Category:Chiral polyhedra]]