# Fractal Tetrahedra

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??

# Warped polyhedra

So I’ve finally finished the pair of beaded shapes I was working on over the last few months! Here they are – a rhombic hexecontahedron and what is probably best described as a hyperbolic dodecahedron:

So around the start of July I was reading about various polyhedra and I came across a rhombic hexecontahedron (the shape on the right) and realised that I could make one out of warped squares. I then realised that I could do a similar shape using warped hexagons and end up with the shape on the left. This isn’t really a polyhedron as the faces aren’t flat, but it’s similar to a hyperbolic dodecahedron shape, which is also known as spikey, the Mathematica logo (while a hexecontahedron is currently the Wolfram Alpha logo). I used Mathematica a lot when I worked in research, and spikey was one of the first ‘mathematical art’ polyhedra I encountered!

It seems that July was a month for making shapes out of warped squares though! While I was making this I saw Joy Davidson’s 3-star beaded box on facebook, and later saw Kat Oliva’s lovely patchwork rhombic hexecontahedron as well. I also ran across a photo of one on pinterest shortly after I finished it, which turned out to be a pattern by June Huber (Juniper Creek Designs). So it seems that I have just reinvented the wheel on this one!

I really like the hyperbolic dodecahedron, although it was at times challenging to make. I managed to make the tension a little too tight on some of the points and there were a couple of broken beads that had to be fixed by removing a section and repairing it, but I finially managed to finish it last week. I was also worried that it would be very difficult to stitch the last few pieces together, but it turned out to be much easier than I thought it would be (curved beading needles are an awesome invention!).

# Icosahedron

I’m busy making a new geometric sculpture, so I thought I’d take a break and take look back at an older piece. This is an icosahedron I made almost 2 years ago now.

I say icosahedron, but it’s really half way between an icosahedron and a dodecahedron. If you think of the beaded ovals as the edges, then you can find groups of three that make the triangular faces of an icosahedron, but you can also find groups of five that make pentagons and overall look like a dodecahedron. However, I based it on a icosahedron when I was making it, so that’s what I’ll call it!

Unlike my beaded dodecahedron, this piece has a chirality because of how the ovals are arranged to make each face. Chirality just means that a shape looks different to its mirror image. You can see that here – the ovals all point out clockwise around each triangle, but the mirror image would have them all pointing anticlockwise. One day I’ll make the mirror image version to match!

I’d also like to try this pattern as a truncated icosahedron (which is an icosahedron with the vertices cut off, so it’s made up out of pentagons and hexagons – like a football). I think it’d look good as a larger shape, but I’m slightly put off by the 90 edge pieces I’d have to make!

# Dodecahedron

As you may have guessed from the kaleidocycle, I like making geometric shapes out of beads! The first shape I tried was a dodecahedron (or an icosahedron, depending on how you look at it). Here’s a photo of the finished piece.

I made this a few years ago but it’s still one of my favourites. The shape is made up of a lot of individual peyote ovals (like a triangle but with only two points, I learnt about them from Diane Fitzgerald’s book Shaped Beadwork).

If you think of it as a dodecahedron then five of the ovals (or rather, five half-ovals) make up each hexgonal face, while if you think of it as an icosahedron then each oval corresponds to an edge.

Each oval is one of five colours, but they occur in a different order in each hexagonal face. After a bit of reading I found out that this is because the symmetries of a dodecahedron (or icosahedron, they’re essentially the same) are the same as a particular permutation group. (Or in full maths detail: its symmetry group is isomorphic to the alternating group A5, which is the group of all even permutations of a set of 5 elements.) So not only does it look cool it has some pretty neat maths behind it as well!