Introduction to Fractal Geometry 2: Julia Sets and the Mandelbrot Set


The Mandelbrot Set – created with Geogebra

It’s possible, or more like probable, that the most mysterious shape ever to arise from the study of Mathematics is that of the Mandelbrot Set. It’s amazing how such an intriguing shape could arise from purely mathematical pursuits. The shape of the Mandelbrot set (often likened to a Beetle), is highly complex yet it is derived from such a simple equation.

Before I get into the post, it’s important to give huge credit to Professor Lee Stemkoski who works at Adelphi University. He figured out a simple and effective way to paint the Mandelbrot set in Geogebra using a lovely little trick – have ten points trace the fractal instead of one! The pictures in this post are a result of some small tweaks to his method.

Any discussion of the Mandelbrot set should start with Julia Sets…

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New Sunflower Applet: Fibonacci and the Golden Angle

Here’s a new applet I created (simply because I haven’t made one in a while and thought it’d be fun). Look at how different angles result in different sunflower configurations by clicking the picture below.


Why is the golden angle 137.5 degrees? If you’re struggling to figure it out based on the standard derivation of the golden ratio then there’s a few videos on youtube that’ll help.

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Anymore ideas for teaching averages?

I was looking at some TES resources the other day trying to find new ideas for teaching mean from a frequency table. I found a few good ones so this post is just a collection of ideas for teaching averages, both from the TES (slightly extended or modified) and things I’ve picked up over the years. This was originally just supposed to be private blog post but I decided to make it public just to see if anyone else has more ideas to bring the topic to life or make it relevant.

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The Mathematics of Rainbows

I’ve always been pretty fascinated by rainbows – anyone surely has to admit that it’s truely amazing how nature can construct such a beautiful image. After many years of wondering, I finally decided to spend the evening learning about them. The problem is that when I started to watch a few videos and read a few websites, they gave information without full explanations (is there anything more infuriating?). Don’t get me wrong, the videos are great for about 90% of the information but I felt like a bit more digging was needed to fully grasp the mathematics behind them. Thus this post is for all of those people that want to learn about the formation of a rainbow but feel like explanations are missing out some of the vital steps in the process.

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Introduction to Fractal Geometry

At University, I did a full mathematics degree which covered around 36 modules – that’s quite a lot of maths! The weird thing about my degree (which is probably the case with most mathematics degrees?), is that most of the stuff we learnt was pretty darn old! To add some perspective, I think the most recent module was General Relativity which Einstein published in 1916. That’s strange! I know very little about maths which has been discovered/invented in the last 100 years!

With this in mind, I decided to look at some more recent stuff (1960’s/70’s) and bought a book on fractal geometry and chaos (James Gleick, Chaos: Making a New Science). It’s a popular Science book which is great as an introduction to the subject in terms of naming the key players and outlining the big ideas. As someone interested in mathematics (I never call myself a mathematician, as to me, this is someone who has or does contribute to the field), I needed to ‘branch out’ for a more in-depth understanding.

In this post, I’ll try to explain some of the basic mathematical principles behind Fractal Geometry from an IB/A-Level perspective. My knowledge only extends across two books, but hopefully this post will enable you to understand the most interesting and important stuff without spending weeks reading around the subject. I’ve enlisted the help of James Tanton and directed you to the Classic Iterated Functions Systems website to help with some of the explanations.

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Introducing Trigonometry – taking a “digital leaf” out of Fawn Nguyen’s book

If you had to pick the most frustrating part of your Secondary School Maths education, what would it be? For me, it is hands down the trigonometric ratios. You can probably guess how it was taught. The teacher simple wrote SOH CAH TOA on the board, gave some examples and then set us off with work from the textbook.

I remember exactly how I felt at the time. Hold the phone, what are these things sin, cos and tan all about?

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Investigations in Mathematics: A Point for Consideration on the New IB Internal Assessment

Almost all IB mathematics teachers are curious about how well students will take to the new guidelines for IB coursework. There are many aspects of this change which I could talk about at length, but I’ve decided to discuss one part which I feel needs to be considered to help to ensure that the exploration is, as the IB put it, “stimulating and rewarding” for all students involved.

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