Fractals, Mandelbrot and the Koch Snowflake

Fractals aren’t actually on the syllabus – but they do offer quite a good opportunity to look at limits, infinite sequences, complex numbers (eg Julia sets etc), the relationship between maths and art and so on.

This video is a fantastic introduction to fractals – looking at how the Koch snowflake has simultaneously a finite area and an infinite perimeter (interesting to link this to Gabriel’s Horn which has finite volume and infinite surface area – though this is not related to fractals):

Even more amazing, the Koch snowflake has a fractional dimension – more than 1 but less than 2.

PBS Nova have created a really detailed and interesting look at fractals and how they occur in real life:

To introduce students to fractals, you can also use the Sierpinski Triangle – which can be generated quite easily (instructions here)

and while you at it, the video of an ultra-detailed Mandelbrot zoom in is pretty impressive – a form of pictorial infinity:

There’s also the Dragon Curve – which is explained in a Youtube video. This allows students to see the beginnings of a fractal design from simply repeatedly folding a strip of paper.

From this you can start to look at both Julia and Mandelbrot sets. Dan Pearcy has posted a fantastic blogpost on the topic – which explains how the amazing fractal nature of these shapes are generated. There are also some amazing Julia Set generators and Mandelbrot generators on Geogebra.

All of this can lead onto the coastline paradox (or indeed, it might be a good place to start a lesson) – which asks can we ever actually measure the length of a coastline – because the more detail we go into, the longer the perimeter becomes.  A good link to ToK knowledge.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.