Monday, 31 December 2012

How to make a Sierpinski Christmas Tree

This Christmas tree (which is yet to be completed) is inspired by the Sierpinski triangle fractal. The image below shows the progression of the fractal at different levels of complexity.
 The evolution of the Sierpinski triangle
The number of black triangles in each diagram forms the geometrical progression 1, 3, 9, 27, … . It is an example of a fractal, which means that it has a self-similar structure and you can see the same pattern repeated over and over again.
My challenge with Year 10 this year was to take this design into three dimensions and create a fractal Christmas tree. We started with regular tetrahedrons constructed with a pair of compasses and a straight edge. In just an hour the girls managed to create one and a half level 4 fractals each containing 43 = 64 tetrahedrons. They worked really hard and I promised them that somehow I’d get up to the next level of fractal, which has a total of 256 tetrahedrons in it. My Year 12s and 13s helped but I was only just over half way through at the end of the day and I’m still working on it now!
This is the first one that they created:
So, this is how to do it…
1. Create the nets of four tetrahedrons and sellotape or glue them together to make a larger tetrahedron. Do this by taping three tetrahedron together on the base and adding a fourth one above.
SAM_0008_fb     SAM_0006_fb
2. Remember that fractals require a self-similar structure so you now need to make four of these structures and tape together to make the next stage in the fractal.
SAM_0007_fb     SAM_0009_fb
3. Repeat this process as many times as you like. Create four of the current structures and tape them together in exactly the same way.
I’ll post the finished tree sometime in the middle of next week, then I just have to find an appropriate place to keep it safe. It’s going to be beautiful!

[Here's a link to the finished tree - it's huge!]



  2. Really nice. And you've demonstrated and explained the underlying maths very clearly - your students will go on to find recursion and functional programming a doddle!

  3. What weight of green paper did you use?

    1. We just standard 80gsm paper but a weightier paper would certainly make it more robust.


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