Disdyakis triacontahedron#Symmetry

Being a Catalan solid with triangular faces, the disdyakis triacontahedron's three face angles $\backslash alpha\_4,\; \backslash alpha\_6,\; \backslash alpha\_\{10\}$ and common dihedral angle $\backslash theta$ must obey the following constraints analogous to other Catalan solids:

:$\backslash sin(\backslash theta/2)\; =\; \backslash cos(\backslash pi/4)\; /\; \backslash cos(\backslash alpha\_4/2)$

:$\backslash sin(\backslash theta/2)\; =\; \backslash cos(\backslash pi/6)\; /\; \backslash cos(\backslash alpha\_6/2)$

:$\backslash sin(\backslash theta/2)\; =\; \backslash cos(\backslash pi/10)\; /\; \backslash cos(\backslash alpha\_\{10\}/2)$

:$\backslash alpha\_4\; +\; \backslash alpha\_6\; +\; \backslash alpha\_\{10\}\; =\; \backslash pi$

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

:$\backslash alpha\_4\; =\; \backslash arccos\; \backslash left(\backslash frac\{7-4\backslash phi\}\{30\}\; \backslash right)\; \backslash approx\; 88.992^\{\backslash circ\}$

:$\backslash alpha\_6\; =\; \backslash arccos\; \backslash left(\; \backslash frac\{17-4\backslash phi\}\{20\}\; \backslash right)\; \backslash approx\; 58.238^\{\backslash circ\}$

:$\backslash alpha\_\{10\}\; =\; \backslash arccos\; \backslash left(\; \backslash frac\{2+5\backslash phi\}\{12\}\; \backslash right)\; \backslash approx\; 32.770^\{\backslash circ\}$

:$\backslash theta\; =\; \backslash arccos\; \backslash left(\; -\backslash frac\{155\; +\; 48\backslash phi\}\{241\}\; \backslash right)\; \backslash approx\; 164.888^\{\backslash circ\}$

where $\backslash phi\; =\; \backslash frac\{\backslash sqrt\{5\}+1\}\{2\}\; \backslash approx\; 1.618$ 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.

File:Icosahedral reflection domains.png 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

The 62 vertices of a disdyakis triacontahedron are given by:{{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 }}

• Twelve vertices $\backslash left(0,\; \backslash frac\{\backslash pm\; 1\}\{\backslash sqrt\{\backslash phi+2\}\}\; ,\; \backslash frac\{\backslash pm\; \backslash phi\}\{\backslash sqrt\{\backslash phi+2\}\}\; \backslash right)$ and their cyclic permutations,

• Eight vertices $\backslash left(\backslash pm\; R,\; \backslash pm\; R,\; \backslash pm\; R\backslash right)$,

• Twelve vertices $\backslash left(0,\; \backslash pm\; R\backslash phi,\; \backslash pm\; \backslash frac\{R\}\{\backslash phi\}\backslash right)$ and their cyclic permutations,

• Six vertices $\backslash left(\backslash pm\; S,\; 0,\; 0\backslash right)$ and their cyclic permutations.

• Twenty-four vertices $\backslash left(\backslash pm\; \backslash frac\{S\backslash phi\}\{2\},\; \backslash pm\backslash frac\{S\}\{2\},\; \backslash pm\backslash frac\{S\}\{2\backslash phi\}\backslash right)$ and their cyclic permutations,

where

:$R\; =\; \backslash frac\{5\}\{3\backslash phi\backslash sqrt\{\backslash phi+2\}\}\; =\; \backslash frac\{\backslash sqrt\{25\; -\; 10\backslash sqrt\{5\}\}\}\{3\}\; \backslash approx\; 0.5415328270548438$,

:$S\; =\; \backslash frac\{(7\backslash phi\; -\; 6)\; \backslash sqrt\{\backslash phi+2\}\}\{11\}\; =\; \backslash frac\{(2\backslash sqrt\{5\}\; -\; 3)\; \backslash sqrt\{25\; +\; 10\backslash sqrt\{5\}\}\}\{11\}\; \backslash approx\; 0.9210096876986302$, and

:$\backslash phi\; =\; \backslash frac\{1\; +\; \backslash sqrt\{5\}\}\{2\}\; \backslash approx\; 1.618$ 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.

The edges of the polyhedron projected onto a sphere form 15 great circles, and represent all 15 mirror planes of reflective I_h 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" | x120px Disdyakis triacontahedron | x120px Deltoidal hexecontahedron | x120px Rhombic triacontahedron | x125px Dodecahedron | x125px Icosahedron | x125px Pyritohedron

class="wikitable collapsible collapsed" style="text-align: center;" !colspan="4"| Spherical polyhedron

valign=top | 170px | 170px | 170px | 170px

(see rotating model) |colspan="3"| Orthographic projections from 2-, 3- and 5-fold axes

class="wikitable collapsible collapsed" style="text-align: center;" !colspan="5"| Stereographic projections
rowspan="4"| 250px
2-fold ! 3-fold ! 5-fold |rowspan="4"| File:Brückner Vielflache Tafel 02.jpg: Vielecke und Vielflache (1900)]]
x150px | x150px | x150px
x120px | x120px | x120px
colspan="4" style="font-size: small;"| Colored as compound of five octahedra, with 3 great circles for each octahedron. The area in the black circles below corresponds to the frontal hemisphere of the spherical polyhedron.

The disdyakis triacontahedron has three types of vertices which can be centered in orthogonally projection:

class=wikitable |+ Orthogonal projections
align=center !Projective symmetry |[2] |[6] |[10]
Image |100px |100px |100px
Dual image |100px |100px |100px

File:Disdyakis triacontahedron dodecahedral 12-color.png

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. Such a puzzle currently has no satisfactory mechanism. It is the most significant unsolved problem in mechanical puzzles, often called the "big chop" problem.{{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 }}

This shape was used to make 120-sided dice using 3D printing.{{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 }}

Since 2016, the Dice Lab has used the disdyakis triacontahedron to mass-market an injection-moulded 120-sided die.{{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 }} It is claimed that 120 is the largest possible number of faces on a fair die, aside from infinite families (such as right regular prisms, bipyramids, and trapezohedra) that would be impractical in reality due to the tendency to roll for a long time.{{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}}

A disdyakis triacontahedron projected onto a sphere was previously used as the logo for Brilliant, a website containing series of lessons on STEM-related topics.{{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}}

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.{{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}}

class=wikitable align=right width=320 |160px |160px

colspan=2|Polyhedra similar to the disdyakis triacontahedron are duals to the Bowtie icosahedron and dodecahedron, containing extra pairs of triangular faces.[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

{{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 *n32 in orbifold notation, and [n,3] in Coxeter notation.

{{Omnitruncated table}}

{{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)

• {{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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Category:Catalan solids