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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)

xi+1 = 0.85xi +0.04yi

yi+1= -0.04xi+0.85yi+1.6

Transformation 2: (0.07 probability of occurrence)

xi+1 = 0.2xi -0.26yi

yi+1= 0.23xi+0.22yi+1.6

Transformation 3: (0.07 probability of occurrence)

xi+1 = -0.15xi -0.28yi

yi+1= 0.26xi+0.24yi+0.44

Transformation 4: (0.01 probability of occurrence)

xi+1 = 0

yi+1= 0.16yi

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.

xi+1 = 0.85(0)+0.04(0)

yi+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:

xi+1 = 0.2(0) -0.26(1.6)

yi+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:

1) Revision Village

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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.

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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!

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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.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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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.

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IB Maths Resources by Andrew Chambers

All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner). Please contact here for information on webinar training or for business ideas.

IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions 

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

Available to download here.

Modelling Guide


IB Exploration Modelling Guide 

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Modelling Guide includes:

Linear regression and log linearization, quadratic regression and cubic regression, exponential and trigonometric regression, comprehensive technology guide for using Desmos and Tracker.

Available to download here.

Statistics Guide

IB Exploration Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Statistics Guide includes: Pearson’s Product investigation, Chi Squared investigation, Binomial distribution investigation, t-test investigation, sampling techniques, normal distribution investigation and how to effectively use Desmos to represent data.

Available to download here.

IB Revision Notes

IB Revision Notes

Full revision notes for SL Analysis (60 pages), HL Analysis (112 pages) and SL Applications (53 pages).  Beautifully written by an experienced IB Mathematics teacher, and of an exceptionally high quality.  Fully updated for the new syllabus.  A must for all Analysis and Applications students!

Available to download here.

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