dodecahedron (thing)

A fascinating fact about regular dodecahedra that should be in here somewhere: if you draw the diagonals, making a five-pointed star on each of the twelve pentagonal faces, you have just drawn the edges of five cubes.

The following POV-Ray script demonstrates this fact (the cube is formed from the twelve red cylinders, in case you can't see it straight away:)

---

#include "colors.inc"
#include "shapes.inc"

camera {
   location <0, 0, -20>
   up  <0, 1, 0>
   right <4/3, 0, 0>
   look_at <0, 0, 0>
}

light_source {<0, 0, -20> color White}
light_source {<0, 40, -1> color White}

// Ok, this could be done with a lot more mathematical
// elegance, but this is the way I worked it out...

#declare Pent =  // pentagon, with diagonals
union { 
  sphere {
     < 0, 0.85065080834958, 0 >,
     0.031
     pigment { Blue }
  }
  cylinder {
     < 0, 0.85065080834958, 0 >,
     < 0.809016994373713, 0.262865556055407, 0 >,
     0.023 
     pigment { Blue }
  }
  cylinder {
     < 0, 0.85065080834958, 0 >,
     < 0.5, -0.688190960232546, 0 >,
     0.031
     pigment { Red }
  }
  sphere {
     <0.809016994373713, 0.262865556055407, 0 >, 0.031
     pigment { Blue }
  }
  cylinder {
     < 0.809016994373713, 0.262865556055407, 0 >,
     < 0.5, -0.688190960232546, 0 >,
     0.023
     pigment { Blue }
  }
  cylinder {
    < 0.809016994373713, 0.262865556055407, 0 >,
    < -0.5, -0.688190960232546, 0 >,
    0.023
    pigment { Green }
  }
  sphere {
    < 0.5, -0.688190960232546, 0 >, 0.031
    pigment { Blue }
  }
  cylinder {
    < 0.5, -0.688190960232546, 0 >,
    < -0.5, -0.688190960232546, 0 >,
    0.023
    pigment { Blue }
  }
  cylinder { < 0.5, -0.688190960232546, 0 >,
    < -0.809016994373713, 0.262865556055407, 0 >,
    0.023
    pigment { Green }
  }
  sphere {
    < -0.5, -0.688190960232546, 0 >, 0.031
    pigment { Blue }
  }
  cylinder {
    < -0.5, -0.688190960232546, 0 >,
    < -0.809016994373713, 0.262865556055407, 0 >,
    0.023
    pigment { Blue }
  }
  cylinder {
    < -0.5, -0.688190960232546, 0 >,
    < 0, 0.85065080834958, 0 >,
    0.023
    pigment { Green }
  }
  sphere {
    < -0.809016994373713, 0.262865556055407, 0 >, 0.031
    pigment { Blue }
  }
  cylinder {
    < -0.809016994373713, 0.262865556055407, 0 >,
    < 0, 0.85065080834958, 0 >,
    0.023
    pigment { Blue }
  }
  cylinder {
    < -0.809016994373713, 0.262865556055407, 0 >,
    < 0.809016994373713, 0.262865556055407, 0 >,
    0.023
    pigment { Green }
  }
}

#declare Bottom = union {
  object { Pent
    rotate < 0, 0, 72 >
    scale < 8, 8, 8 >
  }
  object { Pent
    scale < 8, 8, 8 >
    rotate < 0, 0, 1 * 72 >
    translate < 0, 8 * 0.688190960232546, 0 > 
    rotate <  90 + 26.56505117708, 0, 0 >
    translate < 0, -8 * 0.688190960232546, 0 > 
  }
  object { Pent
    scale < 8, 8, 8 >
    rotate < 0, 0, 3 * 72 >
    translate < 0, 8 * 0.688190960232546, 0 > 
    rotate <  90 + 26.56505117708, 0, 0 >
    translate < 0, -8 * 0.688190960232546, 0 > 
    rotate < 0 , 0 , 72 >
  }
  object { Pent
    scale < 8, 8, 8 >
    rotate < 0, 0, 2 * 72 >
    translate < 0, 8 * 0.688190960232546, 0 > 
    rotate <  90 + 26.56505117708, 0, 0 >
    translate < 0, -8 * 0.688190960232546, 0 > 
    rotate < 0 , 0 , 2 *  72 >
  }

  object { Pent
    scale < 8, 8, 8 >
    translate < 0, 8 * 0.688190960232546, 0 > 
    rotate <  90 + 26.56505117708, 0, 0 >
    translate < 0, -8 * 0.688190960232546, 0 > 
    rotate < 0 , 0 , 3 * 72  >
  }

  object { Pent
    scale < 8, 8, 8 >
    rotate < 0, 0,  9 * 72 >
    translate < 0, 8 * 0.688190960232546, 0 > 
    rotate <  90 + 26.56505117708, 0, 0 >
    translate < 0, -8 * 0.688190960232546, 0 > 
    rotate < 0 , 0 , 4 * 72  >
  }
}

// now draw the thing!
object { Bottom  
  rotate < 180, 0, 0 > 
  translate <0,0,8*(0.850650808350178 + 1.37638192047016)>
}
object {
        Bottom
}

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For the full group-theoretic explanation of this (and much, much more :-) see:

http://www.maths.uwa.edu.au/Staff/schultz/3P5.2000/3P5.8Icosahedral.html

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Addition, 24 Mar, 2001: As I should probably have guessed, this fact is already documented here. See how to construct a dodecahedron (includes pictures!)