So I was playing around with beaded triangles thinking about making some Sierpiński triangles. These fractals are simple to make – you start with a triangle (the first iteration) and remove an inverted half-size triangle from the centre, leaving three smaller triangles joined together to form the larger one (the second iteration). Then you do the same with each of these three triangles to make the third iteration. Keep doing this and you end up with a series of fractals like this:
I was looking at these and thought – can you do something similar, but with tetrahedra? A quick search told me that yes, you can! It’s called a Sierpiński tetrahedron, or a tetrix. I went and found some beads and started beading straight away!
The first iteration of a tetrix is just a plain tetrahedron:
The matte black beads I used here are some of the first delicas I ever brought, over a decade ago!
The second iteration is where it starts to get more complicated! This is four tetrahedra, half as large as before, assembled to make one larger tetrahedron like this:
Each outer face of the tetrahedron is a Sierpiński triangle!
I was worried that joining the pieces together would be difficult, but I just followed a threadpath as if completing the last row of each missing triangle on the outer faces. This seems to hold the pieces together well, and also means the top piece rests on top of the others at each corner, so it doesn’t collapse.
The third iteration proved to be more of a challenge – at this point my tetrahedra were made up of triangles with only three rows. I split it up into four separate groups of four tiny tetrahedra. Each group is made with one thread, and each face is added by working inwards from an outline connected to the rest, rather than by making each one individually. It was quite tricky to do, and there were a few broken beads – I regret picking a matte finish for the edge beads! – but I managed to stitch it all together in the end. Here’s the completed third iteration:
At this point I had to stop since I couldn’t make the tetrahedra any smaller. Should have started with a larger tetrahedron!
Here’s the completed sequence of beaded fractals all together:
Definitely going to try this again – what’s the largest tetrahedron I can start with??
6 thoughts on “Fractal Tetrahedra”
Wow, love it. But I’m a beginner. I need a pattern
I haven’t put a pattern together for it yet, but I’m thinking about doing so. I’ll update the blog if I do!
These are wonderful! I love Sierpinski triangles and mathematical beading. You totally nailed it!
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I think to do more you can only go bigger. You’ve gone as small as you can go, so now you need to make 15 more of the large size tetrahedra and put them all together. That all being said, I think this is wonderful just the way it is with the 3 iterations.
Yep – I did think about adding to them to make them larger, but decided to leave these ones as they are in the end. (Also I may have run out of beads… :-))