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**Sierpinski Triangle: A picture of infinity**

This pattern of a Sierpinski triangle pictured above was generated by a simple iterative program. I made it by modifying the code previously used to plot the Barnsley Fern. You can run the code I used on repl.it. What we are seeing is the result of 30,000 iterations of a simple algorithm. The algorithm is as follows:

**Transformation 1:**

x_{i+1} = 0.5x_{i}

y_{i+1}= 0.5y_{i}

**Transformation 2:**

x_{i+1} = 0.5x_{i} + 0.5

y_{i+1}= 0.5y_{i}+0.5

**Transformation 3:**

x_{i+1} = 0.5x_{i} +1

y_{i+1}= 0.5y_{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 1-3 and assign 1 for transformation 1, 2 for transformation 2, and 3 for transformation 3. Say I generate the number 2 – therefore I will apply transformation 2.

x_{i+1} = 0.5(0) + 0.5

y_{i+1}= 0.5(0)+0.5

and my new coordinate is (0.5,0.5). I mark this on my graph.

I then repeat this process – say this time I generate the number 3. This tells me to do transformation 3. So:

x_{i+1} = 0.5(0.5) +1

y_{i+1}= 0.5(0.5)

and my new coordinate is (1.25, 0.25). I mark this on my graph and carry on again. The graph above was generated with 30,000 iterations.

**Altering the algorithm**

We can alter the algorithm so that we replace all the 0.5 coefficients of x and y with another number, *a*.

a = 0.3 has disconnected triangles:

When a = 0.7 we still have a triangle:

By a = 0.9 the triangle is starting to degenerate

By a = 0.99 we start to see the emergence of a line “tail”

By a = 0.999 we see the line dominate.

And when a = 1 we then get a straight line:

When a is greater than 1 the coordinates quickly become extremely large and so the scale required to plot points means the disconnected points are not visible.

If I alternatively alter transformations 2 and 3 so that I add b for transformation 2 and 2b for transformation 3 (rather than 0.5 and 1 respectively) then we can see we simply change the scale of the triangle.

When b = 10 we can see the triangle width is now 40 (we changed b from 0.5 to 10 and so made the triangle 20 times bigger in length):

**Fractal mathematics**

This triangle 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. Amazingly fractal patterns don’t fit into our usual understanding of 1 dimensional, 2 dimensional, 3 dimensional space. Fractals can instead be thought of as having fractional dimensions.

The Hausdorff dimension is a measure of the “roughness” or “crinkley-ness” of a fractal. It’s given by the formula:

D = log(N)/log(S)

For the Sierpinski triangle, doubling the size (i.e S = 2), creates 3 copies of itself (i.e N =3)

This gives:

D = log(3)/log(2)

Which gives a fractal dimension of about 1.59. This means it has a higher dimension than a line, but a lower dimension than a 2 dimensional shape.

**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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