If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

**The Barnsley Fern: Mathematical Art**

This pattern of a fern pictured above was generated by a simple iterative program designed by mathematician Michael Barnsely. I downloaded the Python code from the excellent Tutorialspoint and then modified it slightly to run on repl.it. What we are seeing is the result of 40,000 individual points – each plotted according to a simple algorithm. The algorithm is as follows:

**Transformation 1:** (0.85 probability of occurrence)

x_{i+1} = 0.85x_{i} +0.04y_{i}

y_{i+1}= -0.04x_{i}+0.85y_{i}+1.6

**Transformation 2:** (0.07 probability of occurrence)

x_{i+1} = 0.2x_{i} -0.26y_{i}

y_{i+1}= 0.23x_{i}+0.22y_{i}+1.6

**Transformation 3:** (0.07 probability of occurrence)

x_{i+1} = -0.15x_{i} -0.28y_{i}

y_{i+1}= 0.26x_{i}+0.24y_{i}+0.44

**Transformation 4:** (0.01 probability of occurrence)

x_{i+1} = 0

y_{i+1}= 0.16y_{i}

So, I start with (0,0) and then use a random number generator to decide which transformation to use. I can run a generator from 1-100 and assign 1-85 for transformation 1, 86-92 to transformation 2, 93-99 for transformation 3 and 100 for transformation 4. Say I generate the number 36 – therefore I will apply transformation 1.

x_{i+1} = 0.85(0)+0.04(0)

y_{i+1}= -0.04(0)+0.85(0)+1.6

and my new coordinate is (0,1.6). I mark this on my graph.

I then repeat this process – say this time I generate the number 90. This tells me to do transformation 2. So:

x_{i+1} = 0.2(0) -0.26(1.6)

y_{i+1}= 0.23(0)+0.22(1.6)+1.6

and my new coordinate is (-0.416, 1.952). I mark this on my graph and carry on again. The graph above was generated with 40,000 iterations – let’s see how it develops over time:

**1000 iterations:**

**10,000 iterations:**

**100,000 iterations:**

**500,000 iterations:**

If we want to understand what is happening here we can think of each transformation as responsible for a different part of our fern. Transformation 1 is most likely and therefore this fills in the smaller leaflets. Transformations 2 and 3 fill in the bottom left and right leaflet (respectively) and transformation 4 fills in the stem.

It’s quite amazing to think that a simple computer program can create what looks like art – or indeed that is can replicate what we see in nature so well. This fern is an example of a self-similar pattern – i.e one which will look the same at different scales. You could zoom into a detailed picture and see the same patterns repeating. You might want to explore the idea of fractals in delving into this topic in more detail.

**Changing the iterations**

We can explore what happens when we change the iterations very slightly.

**Christmas tree**

**Crazy spiral**

**Modern art**

You can modify the code to run this here. Have a go!

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

**Essential Resources for IB Teachers**

If you are a **teacher** then please also visit my new site. This has been designed specifically for teachers of mathematics at international schools. The content now includes over **2000 pages of pdf content** for the entire SL and HL Analysis syllabus and also the SL Applications syllabus. Some of the content includes:

**Original pdf worksheets**(with full worked solutions) designed to cover all the syllabus topics. These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.**Original Paper 3 investigations**(with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.- Over 150 pages of
**Coursework Guides**to introduce students to the essentials behind getting an excellent mark on their exploration coursework. - A large number of
**enrichment activities**such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more. I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

**Essential Resources for both IB teachers and IB students**

1) Exploration Guides and Paper 3 Resources

I’ve put together a **168 page** Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission. Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator! I have also made **Paper 3 packs** for HL Analysis and also Applications students to help prepare for their Paper 3 exams. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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June 27, 2021 at 6:40 pm

YogeetaExcellent exploration