There’s a certain beauty in the inevitability of the 5 Platonic solids – we can say with certainty that they are all of the possible regular tilings of the sphere, and that they are the only possibilities.
There are also many neat and beautiful relationships between them (such as the dualities, the space filling packing of cubes, or of tetrahedra and octahedra).
So far, so tidy, but let’s take a look at the tetrahedron. Take 2 and join them face to face…take another and join it so they all share an edge…then another…and another…you are back to the beginning – full circle…almost!
If it wasn’t for this funny little gap then you could fit 20 of them together to make an icosahedron – linking the solid with fewest faces with the one with the most faces, making a complete loop. It would also lead to lots of other nice relationships – octahedra would fit together with icosahedra to fill space, and dodecahedra would fill space on their own.
But the reality is they simply do not fit. With physical models you can squeeze them together so it looks like they do*, and Aristotle seems to have believed that they did. It’s an easy mistake to make if you dont have a computer because the difference is so tiny – If the dihedral angle of the tetrahedron was only about 1.4712 degrees (or 0.004 of a circle) bigger then it would fit. Of course in the world of mathematics an almost doesn’t count for much, but it seems odd to me. It looks a bit like a mistake – What is this almost doing in the Platonic world of perfection and exactitude?
*Actually this is what drew my attention to this idea in the first place – In my first year of architecture I was playing around with paper octahedra, joining them in this way, it worked fine for a while but when the packing got beyond a certain size it kept pulling apart. At the time I put it down to the thickness of the paper. It wasnt until I modeled it on the computer that the real reason became clear.