Beaded machines

# Cuboctahedron Prismatic Structure by Sieuwke Bijlsma

I recently wrote about reconfigurable materials – materials or surfaces which can be folded into different shapes in a similar way to kaleidocycles or folding cubes. Diane Fitzgerald posted a challenge in the Johnson Solids Project group on facebook to try making beadwork versions of some of these structures and there have been some amazing responses. I’m really excited to be able to show one of these pieces here, a cuboctahedron prismatic structure made by Sieuwke Bijlsma. It’s absolutely stunning!

This piece is a reconfigurable material originating from the research of Johannes Overvelde. You can find more details on the structure in the paper Rational design of reconfigurable prismatic architected materials (Overvelde et al., 2017, Nature 541, 347). It’s based on a cuboctahedron:

Prisms are then added to each face. This is done by placing squares along each edge like so:

When you replace all the faces on a cuboctahedron with prisms like this you end up with a structure that looks like this:

This structure will now fold into lots of different shapes! In beadwork the joins between squares are hinges like those used in a peyote triangle kaleidocycle.

This structure alone is pretty amazing, but it turns out you can join several of them together to make something even more spectacular! If you join two together you get this:

You can then join two of these compounds together to end up with a ‘square’ of prismatic cuboctahedra:

And then you can join two of these ‘squares’ together to get a ‘cube’ of prismatic cuboctahedra!

This is the amazing structure that Sieuwke has beaded!

Because the individual prismatic cuboctahedra will fold into different shapes, the combination of eight of them also folds into different shapes! You can see this in this video by Johannes Overvelde:

Here’s a view of Sieuwke’s beaded structure folded into a different shape:

Here’s another configuration:

The structure is made up of 384 beaded squares and was made over several months. It’s one of the most beautiful beadwork sculptures I’ve seen, and I’m really grateful to Sieuwke for allowing me to show her work to you all!

The photos in this post are copywrite Sieuwke Bijlsma and are used with permission. The prismatic cuboctahedron structure was designed by Johannes Overvelde. The animations in this post were created with Stella4D Pro.

# Beaded Reconfigurable Materials

Reconfigurable materials are materials without a fixed shape – surfaces with a shape that can be changed to different configurations. They have some similarities to kaleidocycles and folding cubes, as you can see from this video from the Harvard John A Paulson School of Engineering and Applied Sciences:

Here’s another video from Johannes Overvelde, one of the researchers who studies these surfaces:

Diane Fitzgerald recently posted a challenge in the Johnson Solids Project group on facebook to try making beadwork versions of these structures. Lots of people rose to the challenge and before long there were lots of photos of beaded reconfigurable materials!

Here’s one I made in response to the challenge:

This is based on the hexagonal prism unit from the paper Rational design of reconfigurable prismatic architected materials (Overvelde et al., 2017, Nature 541, 347), which you can see in subfigure k in Supplementary Figure 7.

You can see that it follows the outline of a hexagonal prism, with pairs of squares added to each edge. It reconfigures to a lot of different shapes:

It’s interesting to see just how different it can be made to look! However, it is also however very fragile, as the peyote squares put the corner beads under a lot of pressure, so you need to be very very careful with it (I had a sliver of glass ping off one of the beads while folding it into a different shape!).

If you want to try making one of these fragile but interesting shapes, here’s a brief walkthrough of how I made this hexagonal prism unit. I used the same sized squares as in the Beaded Johnson solid project and used size 15 seed beads for the hinges.

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# Decagonal Kaleidocycle Tutorial

Some while ago I made a decagonal kaleidocycle using irregular tetrahedra based on a paper model of a half-closed decagonal kaleidocycle by Gijs Korthals Altes. Because the tetrahedra have different length sides the different faces you see as it turns are all different shapes.

I drafted a tutorial for this a while ago, and have finally got around to finishing it – and here it is!

# Tutorial

This tutorial is also available as a pdf!

This kaleidocycle is made from ten tetrahedrons. Each tetrahedron is made from six peyote ovals. There are two different types of tetrahedra and each of these contains four different types of ovals.

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# Oval Kaleidocycle Tutorial

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!

# Tutorial

This tutorial is also available as a pdf!

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.

# Gyroelongated Square Bipyramid Kaleidocycle

Here’s my latest kaleidocycle!

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:

Put this in the middle of the square bipyramid (octahedron) and you get a gyroelongated square bipyramid:

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!

# Mobius warped square kaleidocycle

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!

Beaded machines

# Trefoil Knot Kaleidocycle

A while ago I found an interesting paper about rotating rings of tetrahedra (aka kaleidocycles) by Jean PedersenÂ¹. 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!

Â¹The paper is “Braided Rotating Rings”, Jean J. Pedersen (The Mathematical Gazette, 62, 1978).

# Decagonal Kaleidocycle

Update: a tutorial for this kaleidocycle is available here!

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!

# Folding cube tutorial

Here it is – a detailed tutorial for the folding cube!

This is the first tutorial I’ve ever written so hopefully it makes sense! Any questions just ask.