R Hansen says: >You seem to claim it is the method. Can you give me an example of how, in your mind, you arrived at the proof you spoke of by "visualizing"?
No, I meant to say, it is *one* of the tools available when working out math problems.
As Pam has mentioned, using visualization this way is a fluid process, not a static picture. Any verbal description is bound to be somewhat dissatisfying, but for what its worth...
For the specific problem, I began by forming several pictures in my mind, and asking myself if those pictures shed any light on the problem. Those pictures were formed, obviously, based on knowledge I had acquired earlier. As I said I formed several pictures, and sort of vaguely noted which ones I thought promising. A second pass I tried to form the pictures even more clearly in my mind, and note more clearly the various bits of information I could "extract" from that pictures. After sometime of this kind of concentration, I had a single, simple, and clear picture selected. I knew intuitively that picture had the key information "close to the surface". But I was satisfied and stopped the process, after forming the clearest and simplest picture in my mind that I could. If I had been asked to articulate on the spot a "proof", I would have had a hard time. It was a very simple matter though, at home, with pencil and paper, to translate the picture into an acceptable algebraic ! form (with a couple key facts from calculus.) Only a few lines long.