``I see a confused mass."These are the words the great French mathematician used to describe his initial thoughts when he proved that there is a prime number greater than 11 . His final mental image he described as ``... a place somehere between the confused mass and the first point". In commenting on this in his fascinating but quirky monograph, he asks ``What may be the use of such a strange and cloudy imagery?".
Jacques Hadamard (1865-1963)
Hadamard was of the opinion that mathematical thought is visual and that words only interfered. And when he inquired into the thought processes of his most distinguished mid-century colleaugues, he discovered that most of them, in some measure, agreed (A notable exception being George Pólya).
For the non-professional, the idea that mathematicians ``see" their ideas may be surprising. However the history of mathematics is marked by many notable developments grounded in the visual. Descartes' introduction of ``cartesian" co-ordinates, for example, is arguably the most important advance in mathematics this millenium. It fundamentally reshaped the way mathematicians thought about mathematics. And precisely because it allowed them to ``see" better mathematically.
Indeed, mathematicians have long been aware of the significance of visualization and made great effort to exploit it. Carl Friedrich Gauss lamented, in a letter to Heinrich Christian Schumacher, how hard it was to draw the pictures required for making accurate conjectures. Gauss, whom many consider the greatest mathematician of all time, wrote
``It still remains true that, with negative theorems such as this, transforming personal convictions into objective ones requires deterringly detailed work. To visualize the whole variety of cases, one would have to display a large number of equations by curves; each curve would have to be drawn by its points, and determining a single point alone requires lengthy computations. You do not see from Fig. 4 in my first paper of 1799 , how much work was required for a proper drawing of that curve."The kind of pictures Gauss was looking for would now take seconds to generate on a computer screen.
Carl Friedrich Gauss (1777-1855)
Newer computational environments have greatly increased the scope for visualizing mathematics. Computer graphics offers magnitudes of improvement in resolution and speed over hand-drawn or mentally conceived images and provides increased utility through color, animation, image processing and user interactivity. And, to some degree, mathematics has evolved to exploit these new tools and techniques. We wish to explore some of the more subtle uses of interactive graphical tools which help us ``see" the mathematics more clearly. In particular, we wish to focus on cases where the right picture suggests the ``right theorem'', or where it indicates structure where none was expected, or which offer the possibility of ``visual proof".
For all the various examples we consider we have developed Internet accessible interfaces. They allow the reader interact and explore the mathematics and possibly even discover new results of their own.