Editing Regular icosahedron (section)

== Construction ==
There are several ways to construct a regular icosahedron:
* The construction started from a [[pentagonal antiprism]] by attaching two [[Pentagonal pyramid|pentagonal pyramids]] with [[Regular polygon|regular faces]] to each of its faces.<ref>{{multiref
 |{{harvnb|Silvester|2001|p=[https://books.google.com/books?id=VtH_QG6scSUC&pg=PA141 140&ndash;141]}}
 |{{harvnb|Cundy|1952}}
}}</ref> Such construction led to the regular icosahedron becoming known for [[composite polyhedron|composite]]; the pyramids are the elementary, meaning they cannot be sliced again into smaller convex polyhedrons. This process of construction is known as the [[gyroelongation]], like other polyhedrons in the family of [[gyroelongated bipyramid]]. {{sfn|Berman|1971}}
[[File:Icosahedron-golden-rectangles.svg|thumb|right|Three mutually perpendicular [[golden ratio]] rectangles, with edges connecting their corners, form a regular icosahedron.]]
* Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint. These twelve vertices describe the three mutually perpendicular planes, with edges drawn between each of them.{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/70/mode/1up?view=theater 70]}}
* The regular icosahedron can also be constructed starting from a [[regular octahedron]]. All triangular faces of a regular octahedron are breaking, twisting at a certain angle, and filling up with other equilateral triangles. This process is known as [[Snub (geometry)|snub]], and the regular icosahedron is also known as '''snub octahedron'''.{{sfn|Kappraff|1991|p=[https://books.google.com/books?id=tz76s0ZGFiQC&pg=PA475 475]}}
* One possible system of [[Cartesian coordinate]] for the vertices of a regular icosahedron, given the edge length 2, is: <math display="block"> \left(0, \pm 1, \pm \varphi \right), \left(\pm 1, \pm \varphi, 0 \right), \left(\pm \varphi, 0, \pm 1 \right), </math> where <math>\varphi = (1 + \sqrt{5})/2 </math> denotes the [[golden ratio]].{{sfn|Steeb|Hardy|Tanski|2012|p=[https://books.google.com/books?id=UdI7DQAAQBAJ&pg=PA211 211]}}
By the constructions above, the regular icosahedron is [[Platonic solid]], because it has 20 [[equilateral triangle]]s as it faces. This also results in that regular icosahedron is one of the eight convex [[deltahedron]].<ref>{{multiref
 |{{harvnb|Shavinina|2013|p=[https://books.google.com/books?id=JcPd_JRc4FgC&pg=PA333 333]}}
 |{{harvnb|Cundy|1952}}
}}</ref> It can be unfolded into 44,380 different [[Net (geometry)|net]]s.{{sfn|Dennis|McNair|Woolf|Kauffman|2018|p=[https://books.google.com/books?id=4hVeDwAAQBAJ&pg=PA169 169]}}