By Changemaker Volunteer, Pieter van Staalduinen
Wait! Don’t go just yet! We promise you this math can be done entirely without numbers.
Fractals are simply shapes that are infinitely complex and self-similar. This means that as you “zoom in” on them, they stay as complicated as they were when seen from afar, while still resembling themselves. There are lots of examples of these in nature.
This is the Ganges basin, and we can see that the water (white-ish) runs from small streams to the large body on the right of the image. If we were to zoom in in real time, however, we would see that each of these rivers is, itself, fed by smaller waterways. Those waterways, also, are fed by smaller ones, and on and on until we watch single drops of water running down a blade of grass...
Fractals are really cool to think about because they let us appreciate how complicated nature is.
Consider a snowflake. Snowflakes also show fractal properties, since we know that they are similarly complex (and look like themselves) if we zoom in. And again, we can zoom all the way down to how the molecules of water are arranged when they crystallize into ice and things are still lined up!
As a cool aside, check out Wilson Bentley’s pictures of snowflakes. He is the first known photographer of snow, and lived in Vermont 1865-1931. His work is absolutely beautiful.
There are many more fractals out in the world - what other examples can you and your child find/think of? We’ve come up with wood grain, leaves, and broccoli as examples.
Even better, fractals can be drawn by hand. They can be very simple, or remarkably complex, but my favourite is called a Koch Snowflake. All you have to do is draw an equilateral triangle (one where all sides are the same length), and then draw another equilateral triangle in the middle third of each side:
If you keep going a while, eventually it looks like this:
And you can even tesselate (tile) them to make them even more fractal! Creating these shapes can be incredibly meditative - it takes time and precision to draw the finer details, and only the tools available really stop the process, since a fractal keeps going forever.
There’s a pretty great YouTube channel that discusses drawing fractals as well, here. Happy fractal finding/drawing/making!
Pieter is one of our Changemaker Volunteers. He is an elementary teacher and math specialist as well as a passionate outdoor enthusiast. He is an avid reader, writer, math geek, and rock climber, and has worked as an outdoor educator at various camps for the past decade. You can find him wandering around outside after the rain, or playing with math problems in the colder months.