Monday Maths: Fractal Art

Music is the pleasure the human mind experiences from counting without being aware that it is counting. ~Gottfried Leibniz

This post is part of my ongoing series dedicated to getting children excited about maths.

This is a lesson about geometry. It’s all about drawing shapes. It’s also about art, and making pretty things!

What is a Fractal?

A fractal is a “self-similar” pattern – that is, a pattern that looks the same from close up as it does from far away. Snowflakes are fractals!

They’re built on the foundations of recursion, which is a fancy way of saying that the result of an equation is plugged back in to the equation. So, if I have the equation 3x + 2, then I can start with x=1 and get the following:

3 x 1 + 2 = 5
3 x 5 + 2 = 17
3 x 17 + 2 = 53
3 x 53 + 2 = 161
And so on.

Depending on the equation that can get very big, very quickly – or very small, very quickly.

The word “fractal” was first used by a French man called Benoit Mandelbrot in 1975, which is why they’re sometimes referred to as Mandelbrot patterns.

That all sounds very boring, doesn’t it? Why should we care?

Aside from the fact that fractals can be found all over nature, and understanding them is really important if you want to try and do any kind of scientific work with the natural world, they can also be used to make really pretty art.

A “Julia Set”
Source: Wikipedia

How do you make Fractal Art?

If you’re avoiding complex maths and computer programs (which I am, because I don’t fancy trying to explain imaginary numbers to Brownies), then this is how it’s done.

First, pick a shape and a fraction. I’ll illustrate with a triangle and a third, but you can use any shape. It doesn’t even have to be a regular shape, but for beginners it’s easier with straight edges. The shape I’m describing is technically called a “Koch Snowflake”.

Draw your first shape pretty big in the middle of a piece of paper.

Now, you need some triangles whole sides are a third of the size of the first one, which you can draw attached to each side of the original one.

Keep doing this, getting a third as big each time, until they are so small that you can’t draw them!

The Koch Snowflake
Source: Wikimedia Commons (CC BY-SA)

The picture above shows them in the middle of each side – if you go for one end, then you get interesting spiral patterns. But you have to be consistent – if you draw one at the far right then they all have to be.

20130902-191058.jpg
A triangle-based fractal

Why not have a go? Experiment with different shapes and sizes and see who can make the prettiest pattern!

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