Fractals Part II

This is the second 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 use repeated transformations of lines to generate fractal curves. The first of these, the Koch snowflake, was first described by Helge von Koch in 1904, and is a fractal curve with dimension ≈ 1.26. (See the discussion on dimension below.)



Fractal Art From Lines
Snowflakes, Trees and Dragons – Iterative Fractal Line Art

The Koch Snowflake

Take a straight line, divide it into three equal segments, and replace the middle segment with two segments of equal length, the first rising and the second falling at 60° so they fit. It is not hard to picture the four line-segment pattern that results. Next, repeat the process on each of the four segments. Still not too hard to imagine or draw, but certainly starting to get a little more interesting. Click on the Demo button below to see this process in action.

Now suppose we start with an equilateral triangle, and carry out this process on each side. The activity below lets us do this easily. Click on the Next button multiple times to see how the change accumulates, leading to a shape that is known as the Koch Snowflake. Press on the image and hold to magnify.

 Curve type: 

Change the curve type and repeat the process to see how small changes to the process have a large effect on the final shape. Here are some interesting questions to consider and things to calculate:

  • How many sides does the shape have after each step?
  • Are there any straight line segments left after an infinite number of steps?
  • Can you derive a formula for the area and perimeter after each step?
  • What do you think the area and perimeter will be after an infinite number of steps?

The Heighway Dragon – A space filling curve

Like the Koch curve, we can understand the generation of the Dragon curve as a transformation where each line segment is replaced by something different – in this case a right angle. Click on the Demo button below to see this process in action.

This curve was discovered in 1966 when NASA physicists John Heighway and William Harter considering the outcome of repeated folding. Expecting some kind of random pattern, they found the process kept inviting further folds. You can try this too by starting with a long strip of paper and viewing from the edge. Fold the strip in half by folding the right half over the left half, and make a sharp crease in the middle. Then repeat this process as many times as you are able. Next unfold the paper, and spread each fold to an angle of exactly 90°. The resulting edge curve is the Heighway dragon.

The activity below lets us carry out this process step by step. Click on the Next to see this process in action. Press on the image and hold to magnify.

The activity below lets us carry out this process step by step. Click on the Next to see this process in action. Amazingly, the resulting curve will never cross itself, and 4 copies will fit together exactly to fill a two-dimensional region (click here to see how). The Heighway Dragon is a line with fractal dimension 2. Press on the image and hold to magnify.

Binary Fractal Tree

An alternative way to make fractals from lines is by recursively constructing a binary tree. Specifically, at each step the same simple branching process is applied, but on a smaller scale. You can use the activity below to explore this process. Drag the tops of the branches of the base Y to change the branching shape, and then increase the recursion depth using the slider.



Recursion depth:    

  Colours:

If you press the Symmetric button and turn symmetry off, you can control each branch independently. Press and hold to zoom.

  • See if you can predict the final shape from just the shape of the base.
  • How many end points are there after each step?
  • How many lines are there in total after each step?

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.

Related...

Coding Corner

Click here to use computer code to calculate the perimeter and area for each step of the Koch snowflake construction.