zometool

Zometool is a fantastic product for investigating shapes.  It can be used for anything from investigating simple 2D geometry up to 4-D geometry.

On the right are pictures of me, my colleagues and my students all having loads of fun with it.  It is so easy to use - students can make complex 3D shapes within an hour with little or no instruction.

There is a wealth of free information about Zometool on the web, and lots of ideas for lessons here.  George Hart has written a great book full of ideas how to use it called Zometool Geometry and has other information on his website here.  

some of the maths behind zometool

There are three different coloured connectors; red, yellow and blue of slightly different lengths and cross-sections, and little rhombicosidodecahedron balls to connect them onto.

After a bit of playing, I worked out how to fit them together to make 2D and then 3D shapes.  This is how I would recommend introducing them to students; just let them play and see if they can construct very simple shapes. Before long I was constructing various polyhedra.  

Then my inquisitive nature got the better of me, and I sat down and worked out the significance of the difference in lengths of the three different coloured rods.
It turns out that if you use the blue ones as the side lengths of regular polyhedra, the red and yellow ones are designed to be the radii of the circum-sphere of an icosahedron and dodecahedron respectively!  

I decided to work out the radii of these two polyhedra for a side of unit length - the results are on the right.  Even more exciting is the fact that the first fraction on the right is equivalent to sin72, and that the second one is phi * root(3) / 2, or phi * sin60.

Phi is of course the golden ratio.  Both shapes contain hidden 'golden rectangles'.  Below is a slide show of the construction of the two polyhedra with comments to help you see these golden rectangles.  Once you find them, it is easy to derive the two formulae on the right! (click to play)
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radius of the circumsphere of an icosahedron with side length = 1
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radius of circumsphere of a dodecahedron with side length = 1