Developer: RhinoScript
Summary: Demonstrates one way of creating a Icosahedron.

Can Rhino create Platonic solids, such as an Icosahedron?

Rhino does not have a command that will create an Icosahedron. But, you can write a script that will, using RhinoScript.

The following sample script will create a regular Icosahedron composed of twenty triangular faces, with five meeting at each of the twelve vertices.

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 ' Icosahedron.rvb -- September 2009
 ' If this code works, it was written by Dale Fugier.
 ' If not, I don't know who wrote it.
 ' Works with Rhino 4.0.

 Option Explicit
 
 ' Creates a icosahedron
 '   Vertices: 12
 '   Edges: 30
 '   Faces: 20
 '   Edges per face: 3
 '   Edges per vertex: 5
 '   Sin of angle at edge: 2 / 3
 '   Surface area: 5 * sqrt(3) * edgelength^2
 '   Volume: 5 * (3 + sqrt(5)) / 12 * edgelength^3
 '   Circumscribed radius: sqrt(10 + 2 * sqrt(5)) / 4 * edgelength
 '   Inscribed radius: sqrt(42 + 18 * sqrt(5)) / 12 * edgelength
 '   Coordinates: a and b, where:
 '     a = 1 / 2 and b = 1 / (2 * phi)
 '     phi is the golden ratio = (1 + sqrt(5)) / 2
 Sub Icosahedron()
 
   ' Declare local variables
   Dim radius, center
   Dim sqr5, phi, ratio, a, b
   Dim v(11), s(19)
 
   ' Prompt for center point
   center = Rhino.GetPoint("Center of icosahedral")
   If IsNull(center) Then Exit Sub
 
   ' Prompt for radius  
   radius = Rhino.GetDistance(center, 1.0, "Radius")
   If IsNull(radius) Then Exit Sub
 
   ' This will make the script run faster  
   Call Rhino.EnableRedraw(False)
 
   ' Phi - the square root of 5 plus 1 divided by 2
   sqr5 = Sqr(5.0)
   phi = (1.0 + Sqr5) * 0.5
   ' Golden ratio - the ratio of edgelength to radius 
   ratio = Sqr(10.0 + (2.0 * sqr5)) / (4.0 * phi)
   a = (radius / ratio) * 0.5
   b = (radius / ratio) / (2.0 * phi)
 
   ' Define the icosahedron's 12 vertices
   v(0)  = Rhino.PointAdd(center, Array( 0,  b, -a))
   v(1)  = Rhino.PointAdd(center, Array( b,  a,  0))
   v(2)  = Rhino.PointAdd(center, Array(-b,  a,  0))
   v(3)  = Rhino.PointAdd(center, Array( 0,  b,  a))
   v(4)  = Rhino.PointAdd(center, Array( 0, -b,  a))
   v(5)  = Rhino.PointAdd(center, Array(-a,  0,  b))
   v(6)  = Rhino.PointAdd(center, Array( 0, -b, -a))
   v(7)  = Rhino.PointAdd(center, Array( a,  0, -b))
   v(8)  = Rhino.PointAdd(center, Array( a,  0,  b))
   v(9)  = Rhino.PointAdd(center, Array(-a,  0, -b))
   v(10) = Rhino.PointAdd(center, Array( b, -a,  0))
   v(11) = Rhino.PointAdd(center, Array(-b, -a,  0))
 
   ' Create the icosahedron's 20 triangular faces
   s(0)  = Rhino.AddSrfPt(Array(v(0), v(1), v(2)))
   s(1)  = Rhino.AddSrfPt(Array(v(3), v(2), v(1)))
   s(2)  = Rhino.AddSrfPt(Array(v(3), v(4), v(5)))
   s(3)  = Rhino.AddSrfPt(Array(v(3), v(8), v(4)))
   s(4)  = Rhino.AddSrfPt(Array(v(0), v(6), v(7)))
   s(5)  = Rhino.AddSrfPt(Array(v(0), v(9), v(6)))
   s(6)  = Rhino.AddSrfPt(Array(v(4), v(10), v(11)))
   s(7)  = Rhino.AddSrfPt(Array(v(6), v(11), v(10)))
   s(8)  = Rhino.AddSrfPt(Array(v(2), v(5), v(9)))
   s(9)  = Rhino.AddSrfPt(Array(v(11), v(9), v(5)))
   s(10) = Rhino.AddSrfPt(Array(v(1), v(7), v(8)))
   s(11) = Rhino.AddSrfPt(Array(v(10), v(8), v(7)))
   s(12) = Rhino.AddSrfPt(Array(v(3), v(5), v(2)))
   s(13) = Rhino.AddSrfPt(Array(v(3), v(1), v(8)))
   s(14) = Rhino.AddSrfPt(Array(v(0), v(2), v(9)))
   s(15) = Rhino.AddSrfPt(Array(v(0), v(7), v(1)))
   s(16) = Rhino.AddSrfPt(Array(v(6), v(9), v(11)))
   s(17) = Rhino.AddSrfPt(Array(v(6), v(10), v(7)))
   s(18) = Rhino.AddSrfPt(Array(v(4), v(11), v(5)))
   s(19) = Rhino.AddSrfPt(Array(v(4), v(8), v(10)))
 
   ' Join all of the faces
   Rhino.UnselectAllObjects()
   Call Rhino.SelectObjects(s)
   Call Rhino.Command("_Join", False)
   Rhino.UnselectAllObjects()
 
   ' Don't forget to do this
   Call Rhino.EnableRedraw(True)
 
 End Sub
 
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 Rhino.AddStartupScript Rhino.LastLoadedScriptFile
 Rhino.AddAlias "Icosahedron", "_NoEcho _-RunScript (Icosahedron)"