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>([[:image:disdyakistriacontahedron.gif|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]]
|-
|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, [[Isohedral figure|isohedral]]
|-
|align=center colspan=2| [[File:Disdyakis 30 net.svg|240px|Disdyakis triacontahedron]]<BR>[[Net (polyhedron)|net]]
|}

In [[geometry]], a '''disdyakis triacontahedron''', '''hexakis icosahedron''', '''decakis dodecahedron''', '''kisrhombic triacontahedron'''<ref>Conway, Symmetries of things, p.284</ref> or '''d120''' is a [[Catalan solid]] with 120 faces and the dual to the [[Archimedean solid|Archimedean]] [[truncated icosidodecahedron]]. As such it is [[Isohedral figure|face-uniform]] but with [[Regular polygon|irregular]] face [[polygon]]s. It slightly resembles an inflated [[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]] of the regular [[dodecahedron]] and [[icosahedron]]. It has the most faces among the Archimedean and Catalan solids, with the [[snub dodecahedron]], with 92 faces, in second place. 

If the [[bipyramid]]s, the [[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]], the edges of a disdyakis triacontahedron define 15 [[great circle]]s. [[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]].

== 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]].

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]] 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">{{Cite web |url=http://dmccooey.com/polyhedra/DisdyakisTriacontahedron.html |title=DisdyakisTriacontahedron |access-date=2023-07-29 |archive-date=2022-02-14 |archive-url=https://web.archive.org/web/20220214093848/http://dmccooey.com/polyhedra/DisdyakisTriacontahedron.html |url-status=live }}</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{1 + \sqrt{5}}{2} \approx 1.618</math> is the [[golden ratio]].

In the above coordinates, the first 12 vertices form a [[regular icosahedron]], the next 20 vertices (those with ''R'') form a [[regular dodecahedron]], and the last 30 vertices (those with ''S'') form an [[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]]s, and represent all 15 mirror planes of reflective ''I<sub>h</sub>'' [[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]] 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="5"| [[Stereographic projection]]s
|-
|rowspan="4"| [[File:Spherical disdyakis triacontahedron as compound of five octahedra.png|250px]]
|-
! 2-fold
! 3-fold
! 5-fold
|rowspan="4"| [[File:Brückner Vielflache Tafel 02.jpg|thumb|300px|The 5-fold projection is the main drawing on the right page.<br><small>[[Max Brückner]]: ''Vielecke und Vielflache'' (1900)</small>]]
|-
| [[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]], 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 puzzle]]s like the [[Rubik's cube]]. Such a puzzle currently has no satisfactory mechanism. It is the most significant unsolved problem in mechanical puzzles, often called the "big chop" problem.<ref>{{Cite web |url=http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1500 |title=Big Chop |access-date=2015-08-24 |archive-date=2022-07-30 |archive-url=https://web.archive.org/web/20220730000809/https://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1500 |url-status=live }}</ref>

This shape was used to make 120-sided dice using 3D printing.<ref>{{Cite web |url=http://www.dicecollector.com/DICEINFO_WHAT_SHAPES_DO_DICE_HAVE.html#D120 |title=Kevin Cook's Dice Collector website: d120 3D printed from Shapeways artist SirisC |access-date=2016-04-07 |archive-date=2022-04-10 |archive-url=https://web.archive.org/web/20220410005801/http://www.dicecollector.com/DICEINFO_WHAT_SHAPES_DO_DICE_HAVE.html#D120 |url-status=live }}</ref>

Since 2016, the Dice Lab has used the disdyakis triacontahedron to mass-market an injection-moulded 120-sided [[Dice | die]].<ref>{{Cite web |url=http://thedicelab.com/d120.html |title=The Dice Lab |access-date=2016-04-07 |archive-date=2016-12-08 |archive-url=https://web.archive.org/web/20161208150928/http://thedicelab.com/d120.html |url-status=dead }}</ref> It is claimed that 120 is the largest possible number of faces on a fair die, aside from infinite families (such as right regular [[Prism (geometry)|prism]]s, [[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 triacontahedron [[Spherical polyhedron|projected onto a sphere]] was previously used as the logo for [[Brilliant.org|Brilliant]], a website containing series of lessons on [[Science, technology, engineering, and mathematics|STEM]]-related topics.<ref>{{Cite web |date= |title=Brilliant |url=https://brilliant.org/ |url-status=dead |archive-url=https://web.archive.org/web/20240101092415/https://brilliant.org/ |archive-date=2024-01-01 |access-date= |website=Brilliant |language=en-us}}</ref>

Because the disdyakis triacontahedron has the highest [[sphericity]] of any [[isohedral figure]], it has been studied for its potential use in constructing [[discrete global grid]] systems for satellite imaging.<ref>{{Cite thesis|last=Hall |first=John |date=2022-04-01 |title=Disdyakis Triacontahedron Discrete Global Grid System |url=https://ucalgary.scholaris.ca/items/1bd11f8c-5a71-48dc-a9a8-b4a8b9021008 |publisher=[[University of Calgary Press]] |access-date=2025-03-30}}</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>
|}
{{Icosahedral truncations}}

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]] with order 2,3,''n'' mirrors at each triangle face vertex. This is *''n''32 in [[orbifold notation]], and [''n'',3] in [[Coxeter notation]].
{{Omnitruncated table}}

==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 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

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

[[Category:Catalan solids]]