dodecahedron

One of the 5 platonic solids.

• 12 pentagonal faces.
• 30 edges.
• 20 vertices.

3 faces (and edges) meet at every vertex.

So the dual polyhedron in the icosahedron.

The dodecahedron is a shape filled with sentiment for me. How many people have nostalgia for geometric solids? Ah, but i do.

When i was Yea high (about 2.5 - 3 feet), my dad got out his drawing board and table saw, and large sheets of plywood and pencils and protractor, and various bits of hardware and carpet scraps, and made a big dodecahedron for me to romp in and around. It dominated the living room in our little West Concord apartment, and is the only thing i really remember, other than losing the bits of games under the dishwasher. It was a hiding place and a romping room, and i could ride my little trike right into the door (really mini-size trike).

It was disassembled when we moved to East Longmeadow, and i don't remember if it was ever reassembled. The pieces sat, stacked, in the basement for ages, being slowly cannibalized for wood over the years. I think there's still one pentagonal piece leaning in a corner of my parents' house, though they've moved three times since then.

So, i may have been the youngest child ever to learn what a dodecahedron is!

It is interesting to note that the Pythagoreans thought that knowledge of the dodecahedron was dangerous, something that the general masses should not be given. They even drowned one of their number, Hippasius, for revealing its existence.

From a 1981 edition of Omni magazine, a riddle:

No hard decode - and the answer I'm told has a different face for every letter of its name.

The answer is a dodecahedron. The first part of the riddle is an anagram. I used this a few times back when I was playing Dungeons & Dragons.

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
}

---

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

---

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

Do*dec`a*he"dron (?), n. [Gr. ; twelve + seat, bottom, base: cf. F. dod'ecaedre.] Geom. & Crystallog.

A solid having twelve faces.

⇒ The regular dodecahedron is bounded by twelve equal and regular pentagons; the pyritohedron (see Pyritohedron) is related to it; the rhombic dodecahedron is bounded by twelve equal rhombic faces.

 

© Webster 1913.