[nix]

The word "visualize" makes this a somewhat leading question (though personally my process is visual/tactile).

The best book I know of about this is Jacques Hadamard's The Psychology of Invention in the Mathematical Field. He asked this same question of the important mathematicians of his day - and because he was Jacques Hadamard, they answered. Einstein's response in particular is fascinating. Although math is published in symbolic form, nearly all the mathematicians surveyed reported that they thought in images, sounds, or movements. Only Polya believed that his creative process involved symbol manipulation.

Seymour Papert's Mindstorms is good reading too - the Logo turtle was very deliberately designed to transfer physical expertise into programming skill.