Editing Regular icosahedron (section)

== Applications ==
{{multiple image
 | total_width = 300
 | image1 = Twenty-sided die (icosahedron) with faces inscribed with Greek letters MET 10.130.1158 001.jpg
 | caption1 = Twenty-sided dice from Ptolemaic of Egypt, inscribed with Greek letters at the faces
 | image2 = ScatDice.JPG
 | caption2 = The ''Scattergories'' twenty-sided die, excluding the six letters Q, U, V, X, Y, and Z
}}
Dice are the most common objects using different polyhedra, one of them being the regular icosahedron. The twenty-sided die was found in many ancient times. One example is the die from the Ptolemaic of Egypt, which  later used Greek letters inscribed on the faces in the period of Greece and Rome.<ref>{{multiref
 |{{harvnb|Smith|1958|p=[https://books.google.com/books?id=uTytJGnTf1kC&pg=PA295 295]}}
 |{{harvnb|Minas-Nerpel|2007}}
}}</ref>
Another example was found in the treasure of [[Tipu Sultan]], which was made out of gold and with numbers written on each face.{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/4/mode/1up?view=theater 4]}} In several [[roleplaying game]]s, such as ''[[Dungeons & Dragons]]'', the twenty-sided die (labeled as [[Dice#Polyhedral dice|d20]]) is commonly used in determining success or failure of an action. It may be numbered from "0" to "9" twice, in which form it usually serves as a ten-sided die ([[Dice#Non-cubical dice|d10]]); most modern versions are labeled from "1" to "20".<ref>{{cite web |url=https://www.gmdice.com/pages/dungeons-dragons-dice |title=Dungeons & Dragons Dice |website=gmdice.com |access-date=August 20, 2019}}</ref> ''[[Scattergories]]'' is another board game in which the player names the category entires on a card within a given set time. The naming of such categories is initially with the letters contained in every twenty-sided dice.{{sfn|Flanagan|Gregory|2015|p=[https://books.google.com/books?id=hViuEAAAQBAJ&pg=PA85 85]}}

{{multiple image
 | total_width = 350
 | align = right
 | image1 = Circogonia icosahedra.jpg
 | caption1 = The [[radiolarian]] ''Circogonia icosahedra''
 | image2 = Dymaxion projection.png
 | caption2 = [[Dymaxion map]], created by the net of a regular icosahedron
}}
The regular icosahedron may also appear in many fields of science as follows:
* In [[virology]], [[Herpesviridae|herpes virus]] have icosahedral [[capsid|shells]], especially well-known in [[adenovirus]].{{sfn|Gallardo|Pérez-Illana|Martín-González|San Martín|2021}} The outer protein shell of [[HIV]] is enclosed in a regular icosahedron, as is the head of a typical [[myovirus]].{{sfn|Strauss|Strauss|2008|p=35&ndash;62}} Several species of [[radiolarian]]s discovered by [[Ernst Haeckel]], described its shells as the like-shaped various regular polyhedra; one of which is ''Circogonia icosahedra'', whose skeleton is shaped like a regular icosahedron.<ref>{{multiref
 |{{harvnb|Haeckel|1904}}
 |{{harvnb|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/6/mode/1up?view=theater 6]}}
}}</ref>
* In chemistry, the [[closo cluster|closo]]-[[carboranes]] are [[Chemical compound|compounds]] with a shape resembling the regular icosahedron.{{sfn|Spokoyny|2013}} The [[crystal twinning]] with [[icosahedral twins|icosahedral shapes]] also occurs in crystals, especially [[nanoparticle]]s.{{sfn|Hofmeister|2004}} Many [[Crystal structure of boron-rich metal borides|borides]] and [[allotropes of boron]] such as [[Allotropes of boron#α-rhombohedral boron|α-]] and [[Allotropes of boron#β-rhombohedral boron|β-rhombohedral]] contain boron B<sub>12</sub> icosahedron as a basic structure unit.{{sfn|Dronskowski|Kikkawa|Stein|2017|p=[https://books.google.com/books?id=e0VBDwAAQBAJ&pg=PA436 435&ndash;436]}}
* In cartography, [[R. Buckminster Fuller]] used the net of a regular icosahedron to create a map known as [[Dymaxion map]], by subdividing the net into triangles, followed by calculating the grid on the Earth's surface, and transferring the results from the sphere to the polyhedron. This projection was created during the time that Fuller realized that the [[Greenland]] is smaller than [[South America]].{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/7/mode/1up?view=theater 7]}}
* In the [[Thomson problem]], concerning the minimum-energy configuration of <math>n</math> charged particles on a sphere, and for the [[Tammes problem]] of constructing a [[spherical code]] maximizing the smallest distance among the points, the minimum solution known for <math>n = 12</math> places the points at the vertices of a regular icosahedron, [[Circumscribed sphere|inscribed in a sphere]]. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.{{sfn|Whyte|1952}}

{{multiple image
 | image1 = Kepler Icosahedron Water.jpg
 | caption1 = Sketch of a regular icosahedron by Johannes Kepler
 | image2 = Mysterium Cosmographicum solar system model.jpg
 | caption2 = [[Johannes Kepler|Kepler's]] Platonic solid model of the [[Solar System]]
 | align = right
 | total_width = 300
}}
As mentioned above, the regular icosahedron is one of the five [[Platonic solid|Platonic solids]]. The regular polyhedra have been known since antiquity, but are named after [[Plato]] who, in his [[Timaeus (dialogue)|''Timaeus'']] dialogue, identified these with the five [[Classical elements|elements]], whose elementary units were attributed these shapes: [[Fire (classical element)|fire]] (tetrahedron), [[Air (classical element) | air]] (octahedron), [[Water (classical element)|water]] (icosahedron), [[Earth (classical element)|earth]] (cube) and the shape of the universe as a whole (dodecahedron). [[Euclid]]'s [[Euclid's Elements|''Elements'']] defined the Platonic solids and solved the problem of finding the ratio of the circumscribed sphere's diameter to the edge length.{{sfn|Heath|1908|p=262, 478, 480}} Following their identification with the elements by Plato, [[Johannes Kepler]] in his ''[[Harmonices Mundi]]'' sketched each of them, in particular, the regular icosahedron.{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/55 55]}} In his ''[[Mysterium Cosmographicum]]'', he also proposed a model of the [[Solar System]] based on the placement of Platonic solids in a concentric sequence of increasing radius of the inscribed and circumscribed spheres whose radii gave the distance of the six known planets from the common center. The ordering of the solids, from innermost to outermost, consisted of: [[regular octahedron]], regular icosahedron, [[regular dodecahedron]], [[regular tetrahedron]], and [[cube]].{{sfn|Livio|2003|p=147}}