Been nosing around a few maths and physics sites today…
I first read Godel, Escher, Bach at the age of about twelve when I borrowed it from the library back in Hull. I didn’t understand much of it at that age, but G?del’s Incompleteness Theorem blew my mind. It still does. There’s a whole load of explanations, in English, of this deciptively simple theorem. I prefer the one by Rudy Rucker – somewere, I’ve got a copy of Infinity and the Mind that I haven’t read it for at least ten years.
I like the idea of Penrose Tiles – you can tile an infinite surface without the tile pattern ever repeating. You can play with Penrise tiles using this rather smart Java applet. There’s an interesting site about the mathematics and history of tiling: the galleries showing the full set of plane symetry groups are very beautiful.
Now goggle your eyes at the Penrose Tribar.
M-theory is a kind of generalisation of Superstring Theory, which is itself what we gor when someone added extra dimensions to the old idea of String theory (sooo Eighties, darling). Greg Egan‘s bookDiaspora uses this type of physics to make possible a very complex plot. Recommended if you like hard SF.