I'm attempting to read Roger Penrose's 1000 page epic called The Road to Reality. I can follow the maths for the first 120 pages but then I'm into stuff that is new for me: Riemann surfaces. I think we're being softened up for General Relativity. It starts with the idea of an imaginary number. Think about the square root of -1. It can't possibly exist because when you square a positive number you get another positive number; when you square a negative number you also get a positive. Nothing can be squared and end up negative. Ah, but that's just for ordinary numbers. Let's invent a number i, an imaginary number that squares to become -1. Then we can get complex numbers that are a mixture of ordinary numbers and these new imaginary numbers eg 4+i or 7+2i. These then can be plotted on graphs with the ordinary numbers of the x-axis and the imaginary numbers on the y-axis. It's called the complex plane. Riemann had the idea of cutting sections of the plane and gluing them together for the cases where functions lead to more than one answer. An example of this would be an ordinary square root eg square root of 4 can be either 2 or -2. Roger Penrose has a diagram like the picture below suggesting that you can glue these bits of the complex plane together. I couldn't figure it out so I had to have a go. You glue together the opposite sections with the same pattern.
