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Fractals and Dimensions

Fractals are some of the most beautiful and most bizarre geometric shapes. They look the same at various different scales – you can take a small extract of the shape and it looks the same as the entire shape. This curious property is called self-similarity.

To create a fractal, you can start with a simple pattern and repeat it at smaller scales, again and again, forever. In real life, of course, it is impossible to draw fractals with “infinitely small” patterns. However we can draw shapes which look just like fractals. Using mathematics, we can think about the properties a real fractal would have – and these are very surprising.

Here you can see, step by step, how to create two famous fractals: the Sierpinski Gasket and the von Koch Snowflake.

Sierpinski Gasketvon Koch Snowflake
To create the Sierpinski Gasket, start with a triangle and repeatedly cut out the centre of every segment. Notice how, after a while, every smaller triangle looks exactly the same as the whole.To create the von Koch Snowflake you also start with a triangle and repeatedly add a smaller triangle to every segment of its edge. After a while, the edge looks exactly the same at small and large scales.

The name “fractals” is derived from the fact that fractals don’t have a whole number dimension – they have a fractional dimension. Initially this may seem impossible – what do you mean by a dimension like 2.5 – but it becomes clear when we compare fractals with other shapes.

two lines
A line has dimension 1. When scaled by a factor of 2, its length increases by a factor of 2.
two squares
A square has dimension 2. When scaled by a factor of 2, its area increases by a factor of 4.
two cubes
A cube has dimension 3. When scaled by a factor of 2, its volume increases by a factor of 8.
two triangles
When the Sierpinski Gasket is scaled by a factor of 2, its area increases by a factor of 3. Looking at the blue increase factors above, we can deduce it must have a dimension between 1 and 2. There aren’t any integers in between 1 and 2, so the dimension of the Sierpinski Gasket has to be fractional. In fact, one can calculate that it is around 1.585.
Calculating the Dimension of Fractals

To calculate the exact dimension of fractals, we need exponents and logarithms. Whenever we scale an object of dimension d by a factor of x, its content (length, area, …) changes by a factor of xd. This is because the object is scaled by x in d directions.

When we scale the Sierpinski Gasket by a factor of 2, its area triples. Therefore we get the equation 2d = 3. We can solve this equation to find that the dimension of the Sierpinski Gasket is d = log23 = 1.585…

When we scale one edge segment of the von Koch Snowflake by a factor of 3, its length quadruples. Therefore we get the equation 3d = 4. Like before, we can solve this equation to find that the dimension of the von Koch Snowflake is d = log34 = 1.262…

Note that even though they are called fractals, these dimensions are not fractions. They are, in fact, irrational numbers.

Fractals are very popular in mathematical visualisation, because they look very beautiful even though they can be created using simple patterns like the ones above. You can zoom into a fractal, and the patterns and shapes will continue repeating, forever.

The Sierpinski Gasket

One of the examples above was the Sierpinski Gasket consisting of countless triangles. We created it by repeatedly cutting out a triangle in the centre of all the other triangles. However there are many other methods for creating this shape – and here are just a few.

mandelbrot set mandelbrot set

Pascal’s Triangle is a number pyramid in which every number is the sum of the two numbers above. Let’s see what happens if we highlight all the even numbers…

Start

Coming Soon: Chaos Game Animation

In the Chaos Game, we start with an empty triangle and select a random point in the middle. We then choose one of the three vertices of the triangle at random, and mark the point at the centre of the line from the random point to the vertex. Then we repeat the process, starting with that new point…

AUTOMATON RULES
AUTOMATON GRID

The cellular automaton is a grid consisting of black and white squares. We start with one black square in the first row; the squares in all following rows are coloured automatically depending on the three squares immediately above

The eight rules at the top determine what a square (red) will look like, depending on the three squares above. Modify the rules by clicking them, and try to find the set of rules that produces something like the Sierpinski Gasket.

The Mandelbrot Set

One of the most famous and most intriguing fractals is the Mandelbrot Set, named after the French mathematician Benoît Mandelbrot (1924 – 2010). When rotated by 90°, it looks a bit like a person, with head, body and two arms. Here is how you can create it:

  • We start with the plane of complex numbers (you can think about it like the two dimensional coordinate system). Every point on the plane is represented by a different number c, and we repeat the following steps for every single point:
  • First we create an infinite sequence of numbers according to the following pattern: We start with 0. Every new number is the previous number squared, plus c. In mathematical notation, we have a sequence (zn), where zn+1 = zn2 + c.
  • If this sequence of numbers always increases and tends to infinity (it diverges), we colour the point white. However if the sequence does not increase beyond a certain limit (if it is bounded), we colour the point black.

We repeat this process for every point in the coordinate system. The collection of all the black points is the Mandelbrot set. Move the blue pin below to explore what happens at various points:

mandelbrot set

For the point c = 1 we create the following sequence:

02+1=1
12+1=2
22+1=5
52+1=26
    

This sequence will always increase and tends to infinity, so 1 is not part of the Mandelbrot set. The point is coloured white.

A computer can do these computations very quickly for millions of numbers c – like all pixels on a screen. The code required is simple, but the resulting fractal is unbelievable complex.

When Benoît Mandelbrot studied the fractal in the early 1920s, there weren’t any computers to visualise it – in fact he didn’t know exactly what it looked like. But using mathematics he was able to predict its complexity. The first computer generated image of the Mandelbrot set was produced by an IBM supercomputers in 1980; today everybody can do the same calculations on a normal laptop.

Black points in the image below are part of the Mandelbrot set. Coloured areas are not in the Mandelbrot set, and the colour indicates the speed with which the respective sequence of complex numbers diverges (tends to infinity).

Mandelbrot Zoomer

Fractals in Nature and Technology

Fractals clearly can’t appear in Nature – if you would zoom in further and further, you would eventually arrive at molecules and atoms and the pattern has to stop. However, there are many shapes in nature which are very similar to fractals:

Fern

Cauliflower

Blood Vessels

Snowflakes

Coastlines

River Networks

Clouds

Mountain Ranges

These shapes appear to be completely random, but – as with fractals – there is an underlying pattern that determines how the shapes are formed and what they will look like. Mathematics can help us understand the shapes better, and thus has applications in medicine, biology, geology and meteorology.

At the beginning of this article we created a very realistic snowflake using a fractal. Similar processes can be applied to all kinds of computer generated graphics. The water, mountains and clouds in this image are generated entirely by a computer using fractals. These methods can be used, for example, when creating textures for computer games.

The process can also be reversed and used for image compression. Usually, pictures on a computer are saved by remembering the colour of every individual pixel in the image. For large images this can take up a lot of disk space and we want to reduce its size: we want to compress it.

However some parts of the picture may look similar to fractals. Instead of saving the pixels individually, a computer could try to find certain patterns in the picture and only save these patterns – this could save a lot of space. Methods of fractal image compression were developed by Michael Barnsley and Alan Sloan in the 1980s.

Fractals, Contraction Mapping and Compression
By Alexander Shannon, extract from Eureka 62 (© The Archimedeans)

Fractals in 3-Dimensional Space

So far we have only looked at “flat” fractals like the Sierpinski Gasket or the Mandelbrot set. However there is no restriction regarding the number of dimensions a fractal may live in. Here you can see some fractals in 3-dimensional space:

The Menger Sponge

The Mandel Bulb

Wikimedia User Niabot

Wikimedia User Prokofiev