Visualization extends the natural capacity of the mathematician to intuit his subject, to see the entities and objects which are part of his work with the aid of software and hardware. Since the graphics representations are firmly rooted in verifiable algorithms and machines, the images and interfaces may also provide new forms of exposition and possibly even proof. Most important of all, like spacecraft, diving bells and electron microscopes, it takes the human mind to places it has never been and shows it images from a realm yet unseen.