Pictures and mathematics

Ideal mathematical objects exist only in the world of abstract mathematics. For example an ideal line has zero thickness so we can't actually give it a physical representation. Even if we try and imagine it we need to give it some thickness, in which case it ceases to be a line and becomes some sort of very thin rectangle. An ideal line exists only as an abstract mathematical construction defined in the language of mathematics.

Of course we can use the language of mathematics to define much more complicated objects than lines and so it is important to be aware that any computer generated representations of them can only ever be rough approximations, and that these may be further from the original object than a thin rectangle is from a line. For example, just like an ideal line, theoretical objects like the Mandelbrot set can only exist as abstract mathematical constructions which can't be exactly represented on a computer, or anywhere else for that matter. The best we can do with a computer is to produce illustrations that are shadowy glimpses of them, which, even though they may be approximations, still provide a limitless source of fascinating images.

Without wishing to get further entangled in a philosophical discussion, the main point we are trying to make is this: attempting to draw abstract mathematical objects on a computer is inevitably an inaccurate process but it nevertheless turns out to be a good method for creating some very interesting pictures.