Introduction

Hey! My name is Sinead and I’m from Bali, Indonesia. I graduated the IB in November 2019 with a 7 in Maths SL. I got a perfect score on my IA (20/20) where I essentially calculated the fractal dimension of two Indonesian Islands. In this blog post I’m going to be talking about my experence with writing my math IA. You can use this post as inspiration guidance for your writing IA. I will be describing how I came up with my ideas and how I went about the execution and writing process. I would recommend doing a fractal related IA to anyone who is interested in fractals, and is okay with spending a little extra time to study a maths topic that is outside the standard IB math curriculum. In general I found this IA very fun to work on and I was very proud of the final result

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Ideation Process

How I Came Up With the Topic

I started my IA pretty early and so it was before I studied Statistics and Calculus. Had I studied those topics I probably would have done something related to those, but by the time I wanted to start my IA (Term 3 holiday of Year 2) I had not really learned enough from math to do an interesting IA. I always had an interest in fractals as a math concept ever since I learned about them in one random math class I had in primary. Thus I challenged myself to see whether I could create a math IA out of fractals. I probably spent 2 weeks watching random fractal videos online in search of a method that I could use to calculate something related to fractals. This led me to stumble on the concept of the Coastline Paradox. This is the idea that you can’t properly measure the length of a coastline because of how complex it is. I discovered that you can use “the fractal dimension” (a number between 1 and 2) to measure the complexity of a coastline. I then started researching techniques that can be used to calculate the fractal dimension of a coastline. I discovered two methods which can be used: the Box Counting Method and the Hausdorff Method. Finally, to make this relevant to me, I decided that I was going to use these techniques to find the fractal dimension of two Indonesian islands. I went on google maps and just picked two islands which I thought looked complex and settled on Sulawesi and Nusa Tenggara. So yeah, that's basically how I came up with my topic.

#tip Don’t be afraid to do something that is not directly in your syllabus. So long as the maths is commensurate with the level of the course it should work. Plus you will be awarded personal engagement marks for being creative!

Writing Process

The Introduction and Background Information

Because this was a topic that the examiner was unlikely to be familiar with, I made sure that I provided substantial background information so that the examiner could understand what I was doing. I even brought in a huge quote by a famous fractal mathematician to explain part of the concept because it was a complex topic and I thought that I couldn’t explain it any better than the man who wrote the paper on the topic. I spent the first part of the IA exploring the concept of the coastline paradox using Bali, another Indonesian island as an example. Then, in order to really explain what a fractal dimension is, I deconstructed the fractal mathematicians way of defining a dimension. Using this definition, I derived how the idea of the fractal dimension could be defined as an object that sits between 1 and 2 dimensions. As an example, I applied the equations to two famous fractal shapes: The Sierpinkskis triangle and the Koch Snowflake. Finally I explained the connection between fractals and coastlines and how the fractal dimension could be calculated in order to describe the complexity of a coastline. I included a lot more background information in comparison to my other STEM IAs but for me I felt like this was necessary to provide the relevant context needed to understand the mathematical analysis that I was going to perform.

#tip Make sure that if you are bringing in any new concepts to your IA, you make sure that you thoroughly explain them well. Your friend in the same class as you should be able to read your assignment and be able to understand the gist of what you are talking about.  Assume the examiner knows nothing.

Testing the Methods

After researching, I found two methods that could be used to find the fractal dimension. In order to get reflection points, I wanted to have a way to test the validity of my implementation of these methods. Both of the methods require approximations to be made by eye which makes them vulnerable to human error. I decided to test my implementation of these two methods by finding the fractal dimension of an island which has an already known fractal dimension. The plan was thus to compare which method got me closest to the literature value. Then I would use the best method to calculate the fractal dimension of to Indonesian islands.

#tip Reflection should not all have to be done at the end. It's good to be reflecting throughout your IA. The benefit of doing this pretest comparison of methods was that it gave me a chance to show that I was reflecting on my mathematical analysis and not just blinding crunching numbers.

Computing the methods

In order to calculate the fractal dimension, I needed a way to place rulers along the coastlines of islands and put a grid on them. Originally I was thinking about printing paper to do this but then I realised I could do this all on my laptop. And so using photoshop I downloaded photos from the internet of islands and created my own rules and grids to place on the islands. I won’t go into two much detail here for how I did it, but you can watch these two videos to see exactly how I implemented these two methods:

Hausdorf method

Boxcounting method

Common Mistakes with Fractal IAs

Here are a list of mistakes to avoid if you want to do an IA based on fractals

• Weak understanding what the fractal dimension is: This is a tough concept which I’ve seen a lot of students struggle with. Make sure you spend enough time reading so you actually understand it. DO YOUR RESEARCH
• Poor placement of Rulers: You have to place the rules along the coastline as accurately as you can. Try not to take ridiculous short cuts and spend time trying to rearrange the arrows so everything fits.
• Getting the axis labels wrong when plotting the equation: Some people who have done an IA similar to mine have emailed me saying they have problems with the fractal dimension as the number is not between 1 and 2. This is usually because you have the axis labelled wrong.
• Using the border of a country as a coastline. It's called the coastline paradox because it is the COASTLINE which is too complex to measure. A border between two land countries is not a COASTLINE. You can find the fractal dimension of things like cancer cells but a border does not make sense. Only do the part of the country that is touching the sea.

What was good about this IA

I think the reason why I ensured a 7 on my math IA was because of the creativity. It was based on a subject that likely no one had done before and I was very creative with my method and ideation process. I also did it based on a subject that I was passionate about and that meant that I put extra effort into making sure it was perfect.

#protip Try your best to do an IA that is in a topic that you are passionate about! I promise it will pay dividends as you will actually have the motivation to work on it which will mean that you will naturally put more effort into your IA. It can often be a mistake to chose an IA topic just because you think it will give your high marks

Problems with this IA

In table 2, instead of logging N I simply just copy the numbers in log S instead of actually computing log N. So yeah… these values are incorrect, you should really just take the log of the numbers in row “Number of Boxes N”.

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Ideas for Further Investigation

There are a lot of ways this IA can be extended. There are hundreds more islands of which the fractal dimension can be calculated. If you’re tech savvy and know how to code you could potentially write a program that implements the box counting method for you so you don’t have to count all the boxes. One thing that has some research already done is using the fractal dimension to measure the complexity of cells to determine whether they are cancerous or not. These are just a few ideas, as there is a lot of things that can be done with this sort of maths.

If you are seeking direct help with your math IA you can sign up for IA tutoring where we can set you up with one of the tutors who you can chat with directly about how you can take your IA to the next level. We can help with the ideation, writing and draft iteration process. Click the link here for more information :).

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If you want help

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Helpful resources if you want to do a fractal IA:

Here is my Ultimate Guide to the Math IA video which many studnets have DM's me saying that it was helpful.

Here is a video where I am Going Through my Level 7 Math IA

A huge chunk of this IA was initially inspired by this video by 3Blue1Brown.

I used the tutorial in this video to figure out how to calculate the fractal dimension using the Hausdorff method.

This is another video that talks about the Coastline Paradox

Again, here are two videos where I go through two different methods step by step on how to compute the fractal dimension:

Hausdorf method

Boxcounting method:

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