View source for Disdyakis triacontahedron

{{Short description|Catalan solid with 120 faces}}
{| class=wikitable align="right"
!bgcolor=#e7dcc3 colspan=2|Disdyakis triacontahedron
|-
|align=center colspan=2|[[Image:disdyakistriacontahedron.jpg|240px|Disdyakis triacontahedron]]<br>(rotating and [[:File:Disdyakis triacontahedron.stl|3D]] model)
|-
|bgcolor=#e7dcc3|Type||[[Catalan solid|Catalan]]
|-
|bgcolor=#e7dcc3|[[Conway polyhedron notation|Conway notation]]||mD or dbD
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_f1|5|node_f1|3|node_f1}}
|-
|bgcolor=#e7dcc3|Face polygon||[[File:DU28 facets.png|60px]]<BR>[[Scalene triangle|scalene triangle]]
|-
|bgcolor=#e7dcc3|Faces||120
|-
|bgcolor=#e7dcc3|Edges||180
|-
|bgcolor=#e7dcc3|Vertices||62 = 12 + 20 + 30
|-
|bgcolor=#e7dcc3|Face configuration||V4.6.10
|-
|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||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]]||<math>\arccos\left(-\frac{7\phi+10}{5\phi+14}\right) \approx 164.888^{\circ}</math>
|-
|bgcolor=#e7dcc3|[[Dual polyhedron]] || [[File:Polyhedron great rhombi 12-20 max.png|70px]]<br>[[Truncated icosidodecahedron|truncated<br>icosidodecahedron]]
|-
|bgcolor=#e7dcc3|Properties||convex, face-transitive
|-
|align=center colspan=2| [[File:Disdyakis 30 net.svg|240px|Disdyakis triacontahedron]]<BR>[[Net (polyhedron)|net]]
|}

In [[Geometry|geometry]], a '''disdyakis triacontahedron''', '''hexakis icosahedron''', '''decakis dodecahedron''' or '''kisrhombic triacontahedron'''<ref>Conway, Symmetries of things, p.284</ref> is a [[Catalan solid]] with 120 faces and the dual to the [[Archimedean solid|Archimedean]] [[Truncated icosidodecahedron|truncated icosidodecahedron]]. As such it is [[Isohedral figure|face-uniform]] but with [[Regular polygon|irregular]] face [[Polygon|polygon]]s. It slightly resembles an inflated [[Rhombic triacontahedron|rhombic triacontahedron]]: if one replaces each face of the rhombic triacontahedron with a single [[Vertex (geometry)|vertex]] and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the [[Kleetope]] of the rhombic triacontahedron. It is also the [[Barycentric subdivision|barycentric subdivision]] of the regular [[Dodecahedron|dodecahedron]] and [[Icosahedron|icosahedron]]. It has the most faces among the Archimedean and Catalan solids, with the [[Snub dodecahedron|snub dodecahedron]], with 92 faces, in second place. 

If the [[Bipyramid|bipyramid]]s, the [[Gyroelongated bipyramid|gyroelongated bipyramid]]s, and the trapezohedra are excluded, the disdyakis triacontahedron has the most faces of any other strictly convex polyhedron where every [[Face (geometry)|face]] of the polyhedron has the same shape.

[[Projective geometry|Projected]] into a [[Sphere|sphere]], the edges of a disdyakis triacontahedron define 15 [[Great circle|great circle]]s. [[Biography:Buckminster Fuller|Buckminster Fuller]] used these 15 great circles, along with 10 and 6 others in two other polyhedra to define his [[31 great circles of the spherical icosahedron|31 great circles of the spherical icosahedron]].

== Geometry ==
Being a [[Catalan solid]] with triangular faces, the disdyakis triacontahedron's three face angles <math>\alpha_4, \alpha_6, \alpha_{10}</math> and common dihedral angle <math>\theta</math> must obey the following constraints analogous to other Catalan solids:

:<math>\sin(\theta/2) = \cos(\pi/4) / \cos(\alpha_4/2)</math>
:<math>\sin(\theta/2) = \cos(\pi/6) / \cos(\alpha_6/2)</math>
:<math>\sin(\theta/2) = \cos(\pi/10) / \cos(\alpha_{10}/2)</math>
:<math>\alpha_4 + \alpha_6 + \alpha_{10} = \pi</math>

The above four equations are solved simultaneously to get the following face angles and dihedral angle:

:<math>\alpha_4 = \arccos \left(\frac{7-4\phi}{30} \right) \approx 88.992^{\circ}</math>
:<math>\alpha_6 = \arccos \left( \frac{17-4\phi}{20}  \right) \approx 58.238^{\circ}</math>
:<math>\alpha_{10} = \arccos \left( \frac{2+5\phi}{12} \right) \approx 32.770^{\circ}</math>
:<math>\theta = \arccos \left( -\frac{155 + 48\phi}{241} \right) \approx 164.888^{\circ}</math>

where <math>\phi = \frac{\sqrt{5}+1}{2} \approx 1.618</math> is the [[Golden ratio|golden ratio]].

As with all Catalan solids, the dihedral angles at all edges are the same, even though the edges may be of different lengths.

== Cartesian coordinates ==
[[File:Icosahedral reflection domains.png|thumb|The fundamental domains of [[Icosahedral symmetry|icosahedral symmetry]] form a spherical version of a disdyakis triacontahedron. Each triangle can be mapped to another triangle of the same color by means of a 3D rotation alone. Triangles of different colors can be mapped to each other with a reflection or inversion in addition to rotations.]]

[[File:Disdyakis_Triacontahedron.svg|thumb|Disdyakis triacontahedron hulls.]]

The 62 vertices of a disdyakis triacontahedron are given by:<ref name="radii">[http://dmccooey.com/polyhedra/DisdyakisTriacontahedron.html DisdyakisTriacontahedron]</ref>

* Twelve vertices <math>\left(0, \frac{\pm 1}{\sqrt{\phi+2}} , \frac{\pm \phi}{\sqrt{\phi+2}} \right)</math> and their cyclic permutations,

* Eight vertices <math>\left(\pm R, \pm R, \pm R\right)</math>,

* Twelve vertices <math>\left(0, \pm R\phi, \pm \frac{R}{\phi}\right)</math> and their cyclic permutations,

* Six vertices <math>\left(\pm S, 0, 0\right)</math> and their cyclic permutations.

* Twenty-four vertices <math>\left(\pm \frac{S\phi}{2}, \pm\frac{S}{2}, \pm\frac{S}{2\phi}\right)</math> and their cyclic permutations,

where
:<math>R = \frac{5}{3\phi\sqrt{\phi+2}} = \frac{\sqrt{25 - 10\sqrt{5}}}{3} \approx 0.5415328270548438</math>,
:<math>S = \frac{(7\phi - 6) \sqrt{\phi+2}}{11} = \frac{(2\sqrt{5} - 3) \sqrt{25 + 10\sqrt{5}}}{11} \approx 0.9210096876986302</math>, and
:<math>\phi = \frac{\sqrt{5} + 1}{2} \approx 1.618</math> is the [[Golden ratio|golden ratio]].

In the above coordinates, the first 12 vertices form a [[Regular icosahedron|regular icosahedron]], the next 20 vertices (those with ''R'') form a [[Regular dodecahedron|regular dodecahedron]], and the last 30 vertices (those with ''S'') form an [[Icosidodecahedron|icosidodecahedron]].

Normalizing all vertices to the unit sphere gives a ''spherical'' disdyakis triacontahedron, shown in the adjacent figure. This figure also depicts the 120 transformations associated with the full icosahedral group ''Ih''.

==Symmetry==
The edges of the polyhedron projected onto a sphere form 15 [[Great circle|great circle]]s, and represent all 15 mirror planes of reflective ''I<sub>h</sub>'' [[Icosahedral symmetry|icosahedral symmetry]]. Combining pairs of light and dark triangles define the fundamental domains of the nonreflective (''I'') icosahedral symmetry. The edges of a [[Compound of five octahedra|compound of five octahedra]] also represent the 10 mirror planes of icosahedral symmetry.

{|class="wikitable" style="text-align: center;"
|- style="vertical-align: top"
| [[File:Disdyakis 30.png|x120px]]<br>Disdyakis<br>triacontahedron
| [[File:Disdyakis 30 in deltoidal 60.png|x120px]]<br>[[Deltoidal hexecontahedron|Deltoidal<br>hexecontahedron]]
| [[File:Disdyakis 30 in rhombic 30.png|x120px]]<br>[[Rhombic triacontahedron|Rhombic<br>triacontahedron]]
| [[File:Disdyakis 30 in Platonic 12.png|x125px]]<br>[[Regular dodecahedron|Dodecahedron]]
| [[File:Disdyakis 30 in Platonic 20.png|x125px]]<br>[[Regular icosahedron|Icosahedron]]
| [[File:Disdyakis 30 in pyritohedron.png|x125px]]<br>[[Pyritohedron]]
|}

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="4"| [[Spherical polyhedron]]
|- valign=top
| [[File:Disdyakis 30 spherical.png|170px]]
| [[File:Disdyakis 30 spherical from blue.png|170px]]
| [[File:Disdyakis 30 spherical from yellow.png|170px]]
| [[File:Disdyakis 30 spherical from red.png|170px]]
|-
| (see [[:File:Disdyakis 30 spherical.gif|rotating model]])
|colspan="3"| [[Orthographic projection]]s from 2-, 3- and 5-fold axes
|}

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="4"| [[Stereographic projection]]s
|-
|rowspan="4"| [[File:Spherical disdyakis triacontahedron as compound of five octahedra.png|250px]]
|-
! 2-fold
! 3-fold
! 5-fold
|-
| [[File:Disdyakis triacontahedron stereographic d2 colored.svg|x150px]]
| [[File:Disdyakis triacontahedron stereographic d3 colored.svg|x150px]]
| [[File:Disdyakis triacontahedron stereographic d5 colored.svg|x150px]]
|-
| [[File:Disdyakis triacontahedron stereographic d2 colored crop.svg|x120px]]
| [[File:Disdyakis triacontahedron stereographic d3 colored crop.svg|x120px]]
| [[File:Disdyakis triacontahedron stereographic d5 colored crop.svg|x120px]]
|-
|colspan="4" style="font-size: small;"| Colored as [[Compound of five octahedra|compound of five octahedra]], with 3 great circles for each octahedron.<br>The area in the black circles below corresponds to the frontal hemisphere of the spherical polyhedron.
|}

==Orthogonal projections==
The disdyakis triacontahedron has three types of vertices which can be centered in orthogonally projection:
{|class=wikitable
|+ Orthogonal projections
|- align=center
!Projective<br>symmetry
|[2]
|[6]
|[10]
|-
!Image
|[[File:Dual dodecahedron_t012_f4.png|100px]]
|[[File:Dual dodecahedron_t012_A2.png|100px]]
|[[File:Dual dodecahedron_t012_H3.png|100px]]
|-
!Dual<BR>image
|[[File:Dodecahedron_t012_f4.png|100px]]
|[[File:Dodecahedron_t012_A2.png|100px]]
|[[File:Dodecahedron_t012_H3.png|100px]]
|}

== Uses==
[[File:Disdyakis triacontahedron dodecahedral 12-color.png|140px|thumb|''Big Chop'' puzzle]]
The ''disdyakis triacontahedron'', as a regular dodecahedron with pentagons divided into 10 triangles each, is considered the "holy grail" for combination puzzles like the Rubik's cube. This unsolved problem, often called the "big chop" problem, currently has no satisfactory mechanism. It is the most significant unsolved problem in mechanical puzzles.<ref>[http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1500 Big Chop]</ref><br><br>This shape was used to make d120 dice using 3D printing.<ref>[http://www.dicecollector.com/DICEINFO_WHAT_SHAPES_DO_DICE_HAVE.html#D120 Kevin Cook's Dice Collector website: d120 3D printed from Shapeways artist SirisC]</ref> Since 2016, the Dice Lab has used the disdyakis triacontahedron to mass market an injection moulded 120 sided die.<ref>[http://thedicelab.com/d120.html The Dice Lab]</ref> It is claimed that the d120 is the largest number of possible faces on a fair die, aside from infinite families (such as right regular [[Prism (geometry)|prism]]s, [[Bipyramid|bipyramid]]s, and trapezohedra) that would be impractical in reality due to the tendency to roll for a long time.<ref>{{cite web |url=http://nerdist.com/this-d120-is-the-largest-mathematically-fair-die-possible/ |url-status=dead |archive-url=https://web.archive.org/web/20160503114114/http://nerdist.com/this-d120-is-the-largest-mathematically-fair-die-possible/ |archive-date=2016-05-03 |title=This D120 is the Largest Mathematically Fair Die Possible {{!}} Nerdist}}</ref>

A disdyakis tricontahedron [[Spherical polyhedron|projected onto a sphere]] is used as the logo for Brilliant, a website containing series of lessons on [[Engineering:Science, technology, engineering, and mathematics|STEM]]-related topics.<ref>{{Cite web|url=https://brilliant.org/|title=Brilliant {{!}} Learn to think|website=brilliant.org|language=en-us|access-date=2020-02-01}}</ref>

== Related polyhedra and tilings ==
{| class=wikitable align=right width=320
|[[File:Conway_polyhedron_m3I.png|160px]] 
|[[File:Conway_polyhedron_m3D.png|160px]]
|-
|colspan=2|Polyhedra similar to the disdyakis triacontahedron are duals to the Bowtie icosahedron and dodecahedron, containing extra pairs of triangular faces.<ref>[http://www.cgl.uwaterloo.ca/csk/papers/bridges2001.html Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons] {{Webarchive|url=https://web.archive.org/web/20170317145003/http://www.cgl.uwaterloo.ca/csk/papers/bridges2001.html |date=2017-03-17 }} Craig S. Kaplan</ref>
|}

It is topologically related to a polyhedra sequence defined by the face configuration ''V4.6.2n''. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any ''n'' ≥ 7.

With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.

Each face on these domains also corresponds to the fundamental domain of a [[Symmetry group|symmetry group]] with order 2,3,''n'' mirrors at each triangle face vertex. This is *''n''32 in [[Orbifold notation|orbifold notation]], and [''n'',3] in Coxeter notation.

==References==
{{reflist}}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{Citation |last=Wenninger |first=Magnus |title=Dual Models |publisher={{wipe|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 25, Disdyakistriacontahedron)
*''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 285, kisRhombic triacontahedron)

==External links==
*{{Mathworld2 |urlname=DisdyakisTriacontahedron |title=Disdyakis triacontahedron |urlname2=CatalanSolid |title2=Catalan solid}}
*[https://web.archive.org/web/20080804081355/http://polyhedra.org/poly/show/43/hexakis_icosahedron Disdyakis triacontahedron (Hexakis Icosahedron)] – Interactive Polyhedron Model

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