This video of a kaleidocycle made from peyote ovals was the first post on my blog almost four years ago.

The tape on my hands in the video is to cover up scrapes from rowing, not beading the kaleidocycle – and since I can’t go out to row at the moment I took the opportunity instead to finish the tutorial for it that I drafted several years ago to share with you all!

This kaleidocycle is made from six tetrahedrons. Each tetrahedron is made from six peyote ovals. The ovals are all identical apart from the two accent colours used in the pattern. There are then two different combinations of the ovals to form the tetrahedra – pattern 1 and pattern 2, which is a mirror image of pattern 1.

The book “A Mathematical Tapestry” by Peter Hilton and Jean Pedersen has a discussion of the various rotating rings (kaleidocycles) of polyhedra that are possible, including a diagram of one made of 14 hexacaidecadeltahedra – better known as gyroelongated square bipyramids. It was such an intriguing shape I decided to try and construct one from bugle beads. The finished ring is fascinating – in one configuration it’s rigid but in others it’s completely flexible with many degrees of freedom.

It also makes a great bracelet as it will flex enough to fit over your hand but can then be rotated into the rigid configuration to stay on your wrist!

I made the original version with 12mm beads (Matsuno size 5 twisted bugle in Silver-Lined Bronze), but it works with other sizes. The one above is made with 9mm beads (Toho size 3 bugle in Silver-Lined Teal, Opaque Turquoise and Opaque Jet). The bugles just need to be large enough for several thread passes! I use 0.25mm monofilament nylon illusion cord as the thread, which is strong enough not to be damaged by the bugles but thin enough to allow enough passes through each bead.

I did briefly try making a peyote version using triangles (in this case the units are Eva Mari Keiser’s “gyro-eggs”), but unfortunatly it didn’t work very well as the shapes lose their defining sharp geometric shape.

So what is a Gyroelongated Square Bipyramid? It’s two square pyramids (the square bipyramid part) connected with a strip of 8 triangles formed into a ring (the gyroelongated part). Here’s a square pyramid and a square bipyramid (aka an octahedron):

Here’s a strip of 8 triangles which can be made into a ring to make a square antiprism:

It’s an interesting shape! The two pyramids are at angles to each other and you can find pentagons made from 5 triangles at almost every corner.

They can put together into a kaleidocycle by using evenly spaced bugle beads from the middle (the gold ones in the photo above) as shared hinges. These hinges will be at an angle to each other if you look at the shape from the side, rather than parallel. Turns out that this is the critical feature for getting a kaleidocycle to work, and it’s why you end up with sets of mirror polyhedra in a complete cycle.

Below is a tutorial on how to make a four-colour version of this kaleidocycle! Please be careful with it though – remember that it’s made from fragile glass beads which may have sharp edges, so should be treated with care!

A while ago I came across a youtube video with instructions on how to make a Mobius kaleidocycle out of folded paper hypers – a kaleidocycle with a twist! So of course I had to try making one out of warped squares!

It worked! And it’s great fun to play with. It’s just a strip of 7 warped squares made into a Mobius strip (i.e. twisted once before being joined into a ring). But it also cycles!

If the colours look familiar that’s because they’re the CGB prism colourway!

I don’t know if it would work with regular tetrahedra, but I think the warped squares distort more than a beadwork tetrahedron would and allow it to turn.

This is actually the second Mobius kaleidocycle I’ve made – while I was planning out a peyote version of the trefoil knot kaleidocycle I discovered that it also has a twist (which is awesome but did cause a few issues with the pattern I had planned for it!).

It would be interesting to see if other Mobius cycles can work – maybe with a longer strip of squares more than one twist is possible!

A while ago I found an interesting paper about rotating rings of tetrahedra (aka kaleidocycles) by Jean Pedersen¹. Apart from some great instructions on how to make them by braiding two strips of paper together it also mentions that with enough tetrahedra, a kaleidocycle can be tied into a knot and still rotate.

So of course I had to try this! The paper says that the minimum number of tetrahedra required is 22, which is quite a lot. I decided to make them out of bugle beads to test the idea. I made a long strip of them using right angle weave (although in this case the angles aren’t right-angles) and illusion cord . When I had enough tetrahedra I tied the strip into a trefoil knot – this is just an overhand knot with the ends joined together. The completed kaleidocycle looks like a bit like 3 normal kaleidocycle merged together:

Now for the moment of truth – does it rotate properly?

The answer: yes! It took a few tries to work out how to get it to turn properly, but it’s great fun to play with. Here’s a video:

I think this is my favourite kaleidocycle so far! I want to make a peyote tetrahedra version, but the 88 triangles needed might be going to take me a while!

¹The paper is “Braided Rotating Rings”, Jean J. Pedersen (The Mathematical Gazette, 62, 1978).

Last week I finished my second beaded kaleidocycle – a half-closed decagonal kaleidocycle!

It’s made in a similar way to my last kaleidocycle, except that this time the tetrahedra aren’t regular – some of the sides are different lengths. I based the shape of them on a paper model of a half-closed decagonal kaleidocycle from www.korthalsaltes.com – an amazing website with lots of kaleidocycle models!

Here’s a video of it in action:

The “decagonal” part of the name means it’s made of ten tetrahedra, the “half-closed” part means that some the faces meet with no gap in the centre – or at least they’re supposed to! The beaded version ends up with small gaps in the centre of these faces since the beadwork tetrahedra are only an approximation of the exact shapes.

Using tetrahedra with different length sides means that the different faces you see as it turns are all different shapes – which is pretty neat!

The colours didn’t quite turn out how I expected them to, with one side of the kaleidocycle entirely blue – I designed the pattern on just one tetrahedra and didn’t quite manage to predict how it would all fit together. At least now I have a complete model that will help with the next one!

I’m very happy with with it as it is though – I was quite nervous as I was making it that it wouldn’t turn properly, so I’m very happy it rotates as it should! Definitely going to be making more of these!

So I finally managed to finish the new machine I was working on! This time it’s not a kaleidocycle but a folding cube made using cubic right angle weave. Here’s a video:

I can’t find out if these cubes have a technical name, but they seem to generally be known as magic folding cubes. They’re actually quite similar to kaleidocycles, since they’re a ring of eight linked cubes that can be rotated around back to the original starting point. However, they’re also very different since they alternately form two larger cubes during this rotation. I made the faces on each of these bigger cubes distinct – one is just the plain cubic right angle weave surface:

and the other has crystals embedded in it:

Each individual cube is a 4 by 4 block of cubic right angle weave, with a 2 by 2 gap for the crystals on three sides. Each of these cubes are joined to the two neighbouring cubes using modified right angle weave to make a hinge. I used size B nymo, size 11 seed beads and 4mm crystals (although 3mm or a flatter bead might have been better as the 4mm is just slight too big). I also found a curved beading needle a big help for some of the later rows!

If you want to try making one then I recommend making a paper model first for the hinge pattern – there are a lot of websites with instructions for making paper versions and having a model really helped a lot when I was putting it together.

Eventually I’d like to try making one out of 8 stellar octangula (a polyhedron that looks like two intersecting tetrahedra), since they have the same layout of vertices as a cube so could be fitted together in a similar way. Worked out that I’d need to make 192 triangles to do this though – might take a while!

I’ve been making slow progress on beadwork over the summer, but I have had a bit of time to learn more about kaleidocycles and plan out my next project! Unfortunately I haven’t quite finished the current project (more on that soon) but meanwhile I realised I only ever posted a video of my first kaleidocycle and not any photos, so here it is in more detail!

Here’s a video of my first Contemporary Geometric Beadwork machine: a beaded kaleidocycle! I joined the tetrahedrons only at the ends, using jump rings through slightly larger seed beads.