A fractal is a never-ending pattern that repeats itself at different scales. A fractal repeatedly follows a rule and continually reproduces copies of itself in various sizes and/or directions. Fractals are extremely complex, sometimes infinitely complex - meaning you can zoom in and find the same shapes forever. Computer programmes enable us to do this. Below is an example of a Mandelbrot fractal from the iFractal app on my iPhone:
Fractals can be used to describe the many shapes that are found in nature. This includes trees, plants, clouds, coasts, mountains and other rock formations. They continually grow and change in accordance with a mathematical rule. They have a structure of "self-similarity" in that the part looks like the whole. So if you look at one pinna or leaflet of a fern, then it's shape is similar to the frond.
Fractals are surprisingly simple to make. A fractal is made by repeating a simple process again and again. As well as fractals being found in nature, geometric and algebraic fractals can be easily generated and explored too. This can help explain the patterns observed in nature.
The Fractal Foundation has lots of ideas for using fractals. This is aimed at older children but some material is relevant and useful when working with primary aged children. The trick is to keep the activities straightforward. A child with a particular interest in fractals can explore to a complex level using the computer packages recommended on this website to generate their own fractals.
Firstly, children need experience of creating repeating patterns. For example, tiling work or tessellating shapes may have been recently taught and practised. There's lots of examples of this to be found when out and about in the school grounds and local area. Good challenges to undertake include "Find the best example of a tessellating pattern and justify why is it better than the others you found." Encourage children to think of the criteria they will use for making this decision.
The best way to understand the properties and rules of fractals is to make one. This can be undertaken indoors or out. The children should pick a shape and make up a rule for its repetition...
Can you work out the rule? The square is double the size of the square in the bottom left-hand corner. The reverse of this pattern is that the square in the bottom left-hand corner is half the size of the square it occupies.
Once children get the idea, creating simple geometric fractals is a wonderful doodle opportunity! It can also be linked to artwork. Making a class-sized Sierpinski Triangle is one example. Or better still turn it into a whole school project! Here's a small one, made from 30cm and 60cm sticks...
To make expansive fractals, it is better to work outside where there is more space. A very simple fractal dragon can be made by repeating perpendicular lines, e.g.
Challenge children to work in groups or even as a whole class to create a Heighway fractal dragon that can cover the playground. This will require discussion about how to organise the work and assign roles. Groups will also need to think about the materials required. Chalk works well but this activity could use a lot of chalk especially on a rough surface. Paint or even water on a dry playground may be more cost effective. Alternatively, if you have a lot of sticks, then use these. Both Cosy (Tel 01332 370152) and Muddy Faces sell bundles in fixed sizes if you can't locate a free source near you.
Fractals in nature
One way to examine fractals in nature is to compare a branch with the whole of a tree. The pattern of the branches and leaves on the small branch should mimic the overall shape of the whole tree. Again, it is also possible for children experiment with the patterns of trees using sticks.
Once children have grasped this concept of self-similarity, then challenge children to find other examples of nature fractals in the school grounds or neighbourhood. Encourage them to use digital cameras or sketches to demonstrate the similarity of patterns at a micro and macro scale. Using a digital microscope allows for very close up examination too and can help children look for the continuation of a pattern.
Sometimes, the patterns between the large and small scale within a plant do not match up. Ask the children to think up reasons why this might be so. What factors affect the growth of a plant?
There are many apps and programmes available on the Internet which are simple enough for children to use. This gives children an opportunity to make fractals and zoom in to explore the continuous self-replicating patterns. If possible look for one which allows for 3D images to be created from a Mandelbrot or Julia set. This can help children see how fractal patterns are linked to coasts and mountain ranges. This activity would work well with children teaching each other how to use the function.
Another option is to explore fractals on YouTube:
Best Fractal Zoom Ever: 3D graphics not unlike the Grand Canyon
Mandelbrot Fractal Zoom: 2D graphics.
Give children time to explore images of other fractals found in nature. Children can work in groups to put together a poster that demonstrates their understanding of fractals and further information. Here's a lovely example from a group of P7 children at Mile End School, Aberdeen
With these investigations, children can discover and explore at their own level. It gives those who love complex mathematical ideas an opportunity to really play and have fun. At the same time, for other children it allows them to understand that maths really can help us understand the world around us or at least allow us to view it from a different angle (geddit?).