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.
o 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 pices of sticky-tape where two cards meet just to make it a bit more stable, o Now you can make an icosahedron by joining the corners of the rectangles by gluing cotton so that it looks like the picture above.
o 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 icoshedron 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 Coexter's book Introduction to Geometry, 1961, page 163.]
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