zometool


some of the maths behind zometool
There are three different coloured connectors; red, yellow and blue of slightly different lengths and crosssections, 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.
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 circumsphere 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) 