Fractals Part I

This is the first of three activities that explore the construction and visualisation of fractals.

Although arising from simple processes, fractals exhibit infinite complexity, and exist at the nexus of mathematics, nature, and art. No matter how closely you look at a fractal, however much you zoom in, they remain equally complex (i.e. bumpy). They are nowhere smooth!

Two key concepts are:

  • Recursion – i.e. repeating a transformation on its own output. This may remind you of Russian dolls, or the infinite reflections in parallel mirrors, or even the process of revising an essay.
  • Self similarity – the features of the whole object are repeated infinitely often in its parts, at finer and finer scales. At all magnifications, the object exhibits the same complex structure.

In this activity we repeatedly stamp a 2D pattern, getting smaller each time, to generate fractal patterns known as gaskets. Wacław Sierpiński in particular made these shapes famous through his work on them in the early 1900s.



Sierpiński Stamping
Recursive Fractal Art
Click on the cells in the small stamp mask at the bottom right, then use the buttons below to explore the patterns you can make. You can change the stamp pattern at any step.

  Choose a shape:   Choose a base colour:

     

You might like to imagine the kind of objects that a three dimensional version of this process might produce.

About the activity

Although formally defined in only 1975 by Benoit Mandelbrot in his book The Fractal Geometry of Nature, advances in computer graphics have resulted in enormous progress in our understanding of fractals. In many ways they describe the natural world better than simple shapes, and much about the mathematical process of their creation relates directly to natural processes of growth and branching.

Fractals are mathematical entities with non-integer dimension – or, even more bizarre, things like a line that is so wiggly that it actually is two-dimensional!

But what does this actually mean?

Well, for a normal, two-dimensional shape such as a circle or square, doubling the size of the shape means increasing the area by a factor of 22 = 4. The power of 2 comes from the dimension. For example, a 1×1cm square has area 1cm2, and when doubled in size becomes a 2×2cm square with area 4cm2.

For a fractal, doubling the size changes the area differently – you get a 2d change in area when the shape has fractal dimension d – generally not an integer.

Although it is very satisfying to simply draw and admire fractals, it is also a lot of fun to analyse them mathematically. Some things you might like to try include calculating how the area and perimeter change each step of the iteration that produces them. You might find some unexpected results.

Natural fractals include coast lines, river patterns, tree branches, Saturn's rings, snowflakes, and weather patterns. Technology uses self-similarity in things like antenna design (different scales for different frequency reception), and image compression. In art, fractals are used to generate patterns and textures.

The Coastline Paradox illustrates the non-integer dimensional nature of a fractal since the length of a coastline actually depends on the length of the measuring stick used to measure it. The smaller the stick, the longer the measurement of the coastline.