Beaded machines, Tutorials

Trefoil Knot Kaleidocycle Tutorial

Here’s the tutorial for the trefoil knot kaleidocycle I posted a video of a few weeks ago!

This is pretty quick to make, it just takes a couple of hours or so, and it’s also pretty fun to play with when it’s finished!

Materials

The version in this tutorial is made with two colours of bugle beads – one for the hinges and one for the other sides of the tetrahedra. In the original version I used only one colour, but it can be quite hard to see where the hinges are when you’re making the first few tetrahedra!

  • Matsuno 12mm bugle beads:
    • 3g hinge colour, I used gold #634
    • 8g main colour, I used green #647
  • Illusion cord, 0.25mm (0.01″) diameter, clear

Overview

We’re going to make a string of 21 tetrahedra, joined to each other on one bugle bead (which will act as the hinge). We’ll then tie a knot with the string and join the ends together by making the last (22nd) tetrahedron.

The tetrahedra are all made from what I’m going to call tetrahedral right-angle weave, which is exactly like cubic right-angle weave and prismatic weave, except we’re going to be making tetrahedra instead of cubes or prisms.

Step 1 – Making the first tetrahedron

You need about 15′ of thread to make the whole kaleidocycle (that’s about 3 armspans). To make life easier I unreel about half that the spool and then work without cutting it off. When I run out of thread I then unreel the rest needed, cut it off and work from the other end.

Start by stringing 1 gold and 2 green bugles, leaving about a 10″ tail, then form them into a triangle by passing through the first two beads again:

Trefoil_BeadMechanics_1_l

Then string 2 green bugles and pass through the gold bugle again to make a second triangle:

Trefoil_BeadMechanics_3l

Then pass through the green bugle on the other triangle like this:

Trefoil_BeadMechanics_4l

Now we add the final gold bugle to make the tetrahedra. String 1 gold and pass through the green bugles marked 1 and 2 on the image above, and then through the gold bugle again, like this:

Trefoil_BeadMechanics_6l1

Finally, complete the last remaining face of the tetrahedra by passing through the 2 green bugles and the gold again on the other side:

Trefoil_BeadMechanics_6l2

That’s it for the first tetrahedron, now on to the second!

Step 2 – Making the second tetrahedron

The gold bugles are the hinges of the kaleidocycle, and so the next tetrahedron is built from one of the gold bugles in the first. String 2 green bugles and pass back through the gold bugle on the first tetrahedron:

Trefoil_BeadMechanics_7l

String 2 more green bugles and pass through the gold bugle once more:

Trefoil_BeadMechanics_8l

(This last thread pass can get a bit tricky if the gold bugle is a bit narrower than normal – but if you hold the thread with a pair of tweezers that seems to make it easier.)

We now just have to add a gold bugle to the second tetrahedron. Pass through the first green bugle added, string 1 gold bugle, then pass through the top green bugle from the other triangle, like this:

Trefoil_BeadMechanics_9_l

Finally, complete the last face by passing through the gold bugle then the two other green bugles and then the gold bugle again, like so:

Trefoil_BeadMechanics_10l

That’s the second tetrahedron completed!

Trefoil_BeadMechanics_11

Step 3 – Making tetrahedra 3 through 21

The next tetrahedra are all added in the exact same way as the second in Step 2 above:

Trefoil_BeadMechanics_12

Keep going until you have 21 complete tetrahedra. Then add the first two triangles of the last tetrahedra:

Trefoil_BeadMechanics_23

Don’t add the last hinge gold bugle, it will be shared with the first tetrahedron.

You should now have a string of 21 and a half tetrahedra that looks like this:

Trefoil_BeadMechanics_14n

Now the fun part – tying the trefoil knot!

Step 4 – Tying a trefoil knot

A trefoil knot is just a simple overhand knot. Start by laying the string out flat and then moving the working end (the one with the incomplete tetrahedron) over the other end:

Trefoil_BeadMechanics_14l

Now pass the working end under and through the loop:

Trefoil_BeadMechanics_15l

You should now have something that looks like like this:

Trefoil_BeadMechanics_16

We now just need to bring the two ends together like this:

Trefoil_BeadMechanics_16l

I’ve drawn the outline of the knot in blue on this photo. It looks a bit confusing, but don’t worry it’s actually pretty simple when you have the beads in front of you. (It’s just hard to photograph!)

Step 5 – Joining the ends together

Now we just have to join the two ends together. This is done as before, except instead of adding a gold bugle you use the one from the first tetrahedron. Pass through this gold bugle and then remaining two green bugles on one incomplete face on the last tetrahedron:

Trefoil_BeadMechanics_17l

Then pass through the gold bugle again and then green bugles on the other incomplete face of the last tetrahedron:

Trefoil_BeadMechanics_17l2.jpg

When joined it should look like this:

Trefoil_BeadMechanics_18

The kaleidocycle doesn’t have many degrees of freedom when you’ve tied the knot so there isn’t much risk of getting it twisted – there’s only one way it wants to join up at this point! When it’s joined right you it should look like three inter-connected regular kaleidocycles:

Trefoil_BeadMechanics_20

Weave the ends in by passing back along the tetrahedra. I try and join the ends so the tail from the first tetrahedron is woven into the last one, and vice versa, to give a little bit more strength to the join.

You may find that you’re not able to make any more passes through the gold bugles at this point – if so, just zig zag back and forth through the green bugles on one tetrahedron.

Step 6 – Finished!

That’s it! The trefoil knot kaleidocycle is finished – now you just have to learn how to turn the three interlinked parts at the same time! Have fun!

Trefoil_BeadMechanics_21

 

Beaded machines

Trefoil Knot Kaleidocycle

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:

TrefoilKnotKaleidocycle_BeadMechanics_1

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).

Beadwork, Polyhedra, Tutorials

New tutorial!

I’ve finished the tutorial for my first beaded icosahedron – now named Whirlwind! You can find the tutorial in my brand new etsy shop: www.etsy.com/shop/beadmechanics.

whirlwind_beadmechanics_1

I’ve been working on this for a while – it’s been quite a learning experience! The tutorial is 21 pages with more than 60 photos and diagrams – there’s also a net for a paper version of the model you can cut out and make to help with putting the beadwork together!

I’d always intended to make this icosahedron again so I took the opportunity to take photos as I went along so I could write a tutorial. The new version is actually the mirror image of the original – so now I have a matching pair! (Some brief instructions on how to make a second one so you have a matching pair are also included in the tutorial!)

whirwind_beadmechanics_2

Happy beading!

Beaded machines, Beadwork

Decagonal Kaleidocycle

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

decagonal_kaleidocycle_beadmechanics_7

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!

decagonal_kaleidocycle_beadmechanics_11

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!

decagonal_kaleidocycle_beadmechanics_9

Beadwork, Polyhedra

Rhombicosidodecahedron

Rhombi-what? Like a dodecahedron, but with some extra squares and triangles between the pentagons! My beaded version looks like this:

beadmechanics_diamond7

Technically it’s a small rhombicosidodecahedron, since there’s also a great rhombicosidodecahedron, which has hexgaons instead of triangles and decagons (I think that’s the right word for a ten-sided polygon) instead of pentagons.

It’s actually based on my previous icosahedron model, although it ended up being a slightly different shape in the end. It took me a while to work out which polyhedra it corresponded to, but a rhombicosidodecahedron is an expanded icosahedron so that makes sense! Here’s photo of the two together:

beadmechanics_diamond2

Another thing it turned out to be is really difficult to photograph! Not helped either by the lack of sunshine today (why is it always cloudy every time I finish a piece?).

beadmechanics_diamond3

It’s made using 30 individual diamond-shaped pieces. These are made using some CGB techniques – each one is made up of two layers built from an MRAW band, with two side increases on the bottom side and four on the top. It was definitely a bit of a marathon making 30 MRAW bands though!

beadmechanics_diamond4

Hopefully the weather will improve and I’ll be able to get some better photos soon!

Beadwork, Beadwork objects

Spring spirals

It’s been fairly chilly here the last few months and apparently delicas break pretty easily when they’re cold, so I haven’t been able to get much beadwork done. While I wait for spring (maybe further off than I thought as I sit here watching the hail outside!) I’ve been trying to organise my “in-progress” beadwork… a large proportion of which is half-finished test pieces that I don’t want to take apart, but aren’t really anything useful and seem destined to sit in a box forever.

Occasionally however I do manage to make a test piece into an actual object – if only to feel like I’ve achieved something! This one is a small trinket pot I made out of a test piece for an idea about cellini horns a couple of years ago:

beadmechanics_spiral1

The spikes are just Contemporary Geometric Beadwork half-horns – that is, a side incease (wing) that then gets stitched together along its top, instead of decreasing back to the main work like you would for a horn. The spirals all meet at the right place on the join, but don’t quite line up how I’d like where the half-horn meets the rest of the beadwork – if I ever make these again I need to sit down and work out how to get the counts completely correct so there’s a smooth transition to the rest of the work.

beadmechanics_spiral3

It started out like a very small CGB bangle, a plain tubular piece of peyote from a MRAW start. The transition from circular at the base to square at the top is entirely the result of the cellini spirals changing the shape of the beadwork!

I made it into a trinket pot by just adding a few rows of size 11 and 15 seed beads to the MRAW start at the bottom (like the back of a bezel), as well as a row of 15s at the top. I then cut out a piece of card the right size and stuck some grey suede to either side to make the base:

beadmechanics_spiral4

I’m not sure that cellini horns have much of a future as a bangle idea, but a thinner piece could make a pretty interesting pendant!

beadmechanics_spiral5

It is unfortunately a bit lopsided. I probably wouldn’t use matte beads for the sides again, they seem to result in a fabric which doesn’t have much flexibility – a bit of a problem since the cellini horns cause the beadwork to warp significantly! Still, I think it looks pretty neat:

beadmechanics_spiral2

It’s also a nice spring colour – like green shoots emerging from the winter ground!

Beadwork, Polyhedra

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:

beadmechanics_triangle

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:

beadmechanics_tetrix_1

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:

beadmechanics_tetrix_2

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:

beadmechanics_tetrix_3

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:

beadmechanics_tetrix_5

Definitely going to try this again – what’s the largest tetrahedron I can start with??