FRACTALS Going beyond the Mandelbrot Set

The famous Mandelbrot Set likely springs to mind when talking fractals, but it’s just one of a whole group of fractals known as escape-time fractals. As it is there are several categories of fractals that are generated in quite different ways. Escape-time fractals are, perhaps, the most unexpected because, at first sight, it seems barely plausible that applying such a simple formula could produce an image that is visually attractive and literally infinite in the amount of detail it contains. By way of contrast, some of the algorithms used to generate these fractals might seem to be designed specifically to generate a fractal result – but nevertheless we promise to lead you on an astonishing voyage of discovery…

Lost in the Sierpinski Triangle

As fractals are so visual we’re going to leave the maths for later and start by providing a hands-on introduction so you can see fractals on-screen from the very start. The fractal we’re going to look at is called the Sierpinski Triangle and you can generate it with the XaoS software. You can find the Sierpinski Triangle at Fractal > More Formulae > 7 Sierpinski, and investigate it by zooming and panning, using the intuitive interface.

You’ll notice the characteristic nature of a fractal – something that shows self-similarity at all levels of magnification – but in a change from the Mandelbrot Set and other escape-time fractals, that self-similarity is exact. Yes, the Sierpinski Triangle contains nothing more than triangles, at ever smaller scales, going on literally forever.

The real nature of the Sierpinski Triangle is somewhat hidden in , because of its use of colour to make the image more visually attractive. To see more