Modern Maths to Do With Your Primary Schoolers: Fractals

As I talked about previously, I don’t feel that modern maths really enters into the classroom, or general maths curriculum very often. But that doesn’t mean it can’t! In fact, some of this maths is so easy, you can start learning and playing and thinking about it with your preschooler or primary-school aged children.

Here’s some cool maths ideas we’ve done at home.

Fractal Spotting and Trees

Learn about fractals by playing the fractal-spotting game. A fractal is a shape where small parts of the shape look almost or exactly the same as the whole shape. Can you see a fractal?

Examples you can see on a walk: Trees, clouds, mountains, rock-strewn ground, coastlines, ferns, veins on leaves, stars in the sky. Are there any others? Can you make a collection?

In winter you can calculate the Power-law of trees. When they have no leaves – start at the trunk – that’s 1 – how many branches come off the trunk? How many from each branch? How many from that? Are they the same ratio (1:2, 1:3,1:4)? Do the amount of branches at each join go up or down? Are the branches fatter or thinner than the trunk? Is each branch fatter or thinner as it goes further out? Do you think if the branches keep dividing, will it fill the whole space? Is the tree space-filling? (HINT: most are, to maximise the amount of sun each tree can catch.) Will there be gaps?

Explanation

Fractal dimension is slightly different from the 1-2-3 dimensions in most maths. It’s a measure of how much or how little the fractal fills space. And it depends on the amount of branches per larger branch, and the thickness of the branches. So the fractal dimension can be a fraction! Because some fractals never completely fill up their space.

For older kids

Want to calculate the dimension?

Here’s what you need to know:

How many branches does the branch fork have? And how much thinner is each new branch compared to it’s parent branch?

The dimension is calculated using the formula:

Ok it looks complicated, but, if you’ve done logarithms … it becomes:

which you can do on a basic calculator with a few button presses.

For example: The tree has four branches from the trunk. The branches are about 1/3 the thickness of the trunk:
Tree Dimension =  – log(4) / log (1/3)
= (approx)  1.26

Try it with a few trees, or a few drawings of trees. Some trees have the same amount of smaller branches on each branch, some don’t!

What’s going on there? Is it a mistake? Can a tree have more than one number for it’s dimension?

Explanation

Yes! It’s called being a multifractal. As you zoom in (or travel along the branch), the scaling changes, so the dimension changes. This is used in modelling all sorts of things in the real world. One application is in calculating the dimension of how the stars are spaced in the universe*. If you want to know more about the fractal maths of star distribution, start with some Cantor sets.

* Though recent finding suggest that the universe is only fractal-like out to certain scales, rather than a true fractal.

How to Draw a Cool Snowflake

Koch snowflake creation, Wikimedia commons.

You will need a bit of paper for this. If you want, you can do it with a pencil and eraser. Or you can just redraw each new snowflake iteration in a new spot.

Start with a straight line. Now rub out the middle third and put in a triangle (see animation).  Now for each of those lines, rub out the middle third again, and replace it with a triangle that fits the empty space. Now do it again for each of those new lines. And keep doing it for as long as you can!

If you want a full snowflake, instead of starting with a line, start with a triangle.
What happens when you replace the line with a more pointy triangle?  Or put the triangle in a different spot? What about a fatter triangle?

Now let’s try adding a bit of random into the mix.  Start the process again, except this time whenever you add in your new triangle, flip a coin. Heads – triangle points up. Tails – triangle points down. What kid of shape do you get now? Does it look like anything you recognise?

(HINT: Have a look at some pictures natural landscapes. Does it look like a mountain? Does it look like an island? What does the silhouette look like?)

Random Koch, Wikimedia Commons

Explanation

Fractals with random elements are the models used to make realistic landscapes in movies and computer games – and the natural process of growth and erosion of mountains, naturally leads to fractal-like shapes. Random fractals make up a lot of nature!

For older kids

You can keep on getting smaller and smaller lines on a Koch snowflake. What do you think will eventually happen to the size of the lines? How many lines would you have?

Explanation

The straight lines keep getting smaller until they disappear! (Infinite amounts of infinitely small lines) But there are NO smooth parts – it’s all bumps! This means you can’t ‘smooth’ the fractal – which means you can’t differentiate it, or integrate it – calculus doesn’t work! You can’t make ‘averages’ or ‘trends’ in the usual way. It’s continuous, but not differentiable…and it’s infinite in length!

Side-note

This is why most economic theories are bunk – they’re trying to use ‘classical maths’ (yes, even some statistics), on non-classical structures – fractals. Yes, the ups and downs of the stock-market are fractal. The Black Swan by Nassim Nicolas Taleb is also a great introduction to these ideas in economics.

Self-Similarity and Clouds

If you have binoculars – have a look at some fluffy clouds. When you look at it with binoculars, does the edge look the same? Can you see more bumps, or less?

Explanation

This is something called self-similarity – you can’t tell what scale you are looking at, because zoomed in (smaller scales) look a lot like zoomed out scales.

See also information on the Hausdorff-Besicovitch dimension.

Grow a crystal

Diffusion Limited Aggregation, Wikimedia Commons

You will need a seed point, a flat dish, some salt, and a bit of time. And the result should look a little like this image. See also this and this website for instructions.

Explanation

This is something called Diffusion Limited Aggregation – growing branch-like structures from small points. It’s how cracks grow in solid objects like steel, and how cities grow organically at the edges. It is also called ‘edge-growth’.

There are also some cool programs out there for mucking around with ‘growing crystals’. See this and this.

Extra Reading

For more information on fractals, see Benoit Mandelbrot’s famous book that started it all, The Fractal Geometry of Nature.