The Platonic Solids

Things to do

1. I like to have my students chop up a sheet which I call the Platonic solids in 2-D (paper template). Then they use tape to create a set of platonic solids which I permit them to use for their exam.

2. Geomags are great for having students create a set of platonic solids, but the dodecahedron is really tough to do! Worth a try, nonetheless. The students enjoy playing with them, and make sure that you take a tetrahedron and an octahedron and dangle the octa from one of the vertices of the tetra -- really beautiful illustration of how steel-on-steel is about the best friction-reducing scheme possible (railroads!).

   Also remind the students that only three of the five are stable -- only those built with triangles.

3. The following I borrowed from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi3DGeom.html#dodicos He relates this only to the golden rectangle, and to the platonic solids, but by assembling the cards as he does, he's also creating a copy of the Borromean rings!
   1. Here is an interesting way to make a model of an icosahedron based on the three golden rectangles intersecting as in the picture above:
      • Cut out three golden rectangles. One way to do this is to take three postcards or other thin card and cut them so they are 10cm by 16.2cm.
      • In the centre of each, make a cut parallel to the longest side which is as long as the shortest side of a card.
        The three cards will be slotted through these slits to make the arrangement in the picture above. To do this, on one of the cards extend the cut to one of the edges.

             +--------------+ Make  and one   +-------------+
             !              ! two       of    !             !
             !    ======    ! of        these !   ===========
             !              ! these           !             !
             +--------------+                 +-------------+
                   

      • Assemble the cards so that they look like the picture here of the red, green and blue rectangles. [This is a nice little puzzle itself!] You may want to put pieces of sticky-tape where two cards meet just to make it a bit more stable.
      • Now you can make an icosahedron by joining the corners of the rectangles by glueing cotton so that it looks like the picture above.
      • It will be quite delicate, so tape another piece of cotton to one of the short edges of one of the cards and hang it up like a mobile!
   2. If you are good at coordinate geometry or like a challenge, then show that the 12 points of the icosahedron divide the edges of the octahedron in the ratio Phi:1 (or 1:phi if you like) where the octahedron has vertices at:
      ( ±Phi² , 0 0 ), ( 0, ±Phi² , 0 ), ( 0, 0, ±Phi² )
      [from H S M Coxeter's book Introduction to Geometry, 1961, page 163.] http://upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Icosahedron-golden-rectangles.svg/500px-Icosahedron-golden-rectangles.svg.png