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).

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

Beaded machines, Beadwork, Tutorials

Folding cube tutorial

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

beadmechanics_cube_verrier

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

Materials

I used the following beads for the original cube:

  • Size 11 seed beads – about 22g – Miyuki, gunmetal, colour 451
  • 4mm crystals – 24 – Swarovski bicones, blue zircon

You’ll also need:

  • Size B nylon (nymo or s-lon etc.) thread – black
  • Size 12 beading needle
  • Curved beading needle (size 10, but I think that’s the only size you can get)

Cubic Right Angle Weave (CRAW)

The individual cubes are made using cubic right angle weave (CRAW). I’ve written the tutorial assuming familiarity with this stitch, such as how to go round corners and how to build off existing work. If you’re not familiar with it don’t worry – there are many tutorials online! The pattern shouldn’t be too difficult once you master the basics of the stitch.

Modified Right Angle Weave (MRAW)

The hinges on each cube are done with modified right angle weave (MRAW). If you’ve not come across this stitch before you should check out the instructions on the CGB site. They can be found on page 39 of the freely available ‘Basics’ section of CGB volume 1.

Making a model

I found it very helpful to have a model of an unfolding cube to refer to – it makes it much easier to work out which hinge goes where when you have an example right in front of you!

There are several options for making a model. The first is to find 8 small cubes and join them together (see below for the joining pattern). The second is to print out a paper net and make a complete cube all in one go (although this was not the easiest to do!).

Hinge pattern

I’ve used some 20mm wooden cubes and some stickers and tape to put together a model. (Wooden cubes seem pretty easy to get hold off from craft shops or online – or you could also make individual cubes out of paper.)

beadmechanics_cube_model1

The stickers mark the faces where the crystals are going to go – 3 on each cube. Each cube is identical until the hinges are added – this makes everything much easier! They look like this – 3 adjacent faces have stickers/crystals, and the other 3 are blank:

beadmechanics_cube_model2

Here’s how the 8 cubes go together. I’m going to number them 1 to 8 to make the later instructions clearer.

First join cubes 1 and 2 by making a single hinge by sticking tape along the edge as shown (I stuck a piece of tape on both sides of the hinge to make it stronger):

beadmechanics_cube_model3

Then add cube 3 like this, paying close attention to where the stickers are:

beadmechanics_cube_model4

Then add cube 4 on the other side, and fold cubes 3 and 4 in as shown on the right:

beadmechanics_cube_model5

Then add cube 5:

beadmechanics_cube_model6

And cube 6 (still making sure all the stickers are in the right place – although don’t worry if you make a mistake, you can always peel them off and stick them back on!):

beadmechanics_cube_model7

Then add cube 7 like this:

beadmechanics_cube_model8

And finally add cube 8 as shown:

beadmechanics_cube_model9

There’s one more hinge to go – the one between cube 7 and cube 8, which is along the bottom horizontal edge as shown:

beadmechanics_cube_model10

That’s it! You should now have a working model, which folds into two different cubes – one with stickers on all the faces, and one with blank faces:

beadmechanics_cube_model11

If you’ve made a model as above and rotated it around a few times you may notice you can unfold it into a rectangle so it looks like the photos above but the hinges are all in the wrong places. Don’t panic! This is because there are two different places in the complete folding-unfolding cycle where the cube flattens out to a rectangle, and they have the individual cubes in a different order. Just continue folding and unfolding and you’ll get back to where you started.

Now you have a model, on to the beadwork!

The pattern

The pattern is split into 2 parts:

  1. Making the 8 individual CRAW cubes
  2. Adding the hinges to each cube and joining them together

CRAW cubes

Here are the instructions for making each individual cube. You’ll need to make 8 of these.

The individual cubes are made out of cubic right angle weave (CRAW) units.

Start with a comfortable length of thread and a normal size 12 needle – I use about one and a half armspans length of thread at a time, since CRAW uses up the thread fairly fast. When you need to end a thread, just weave the end into the beadwork following the existing CRAW paths until it’s secure, and cut it off. Join the new thread into the beadwork in a similar way.

I’ve used different coloured beads to the original here to make the photos clearer. The original was made entirely using gunmetal coloured beads throughout.

Step 1

Make a 2 by 2 block of CRAW units, leaving enough of a tail to comfortably stitch back in.

(Right click a photo and select ‘view image’ to see a larger version!)

beadmechanics_cube_step1

Step 2

Add another 2 by 2 block on top of this to make a cube.

This will be the centre of the completed cube.

beadmechanics_cube_step2a

This is also a good point to stitch the thread tail in. (It’s also a good idea not to cut it off completely for the time being, just leave it fairly short so it’s out of the way, otherwise you can end up stitching through it and pulling the end back out.)

beadmechanics_cube_step2b

Step 3

We’re now going to add the first ‘frame’ face – one that will eventually have a crystal in the centre. This is done by adding a frame of CRAW to the top of the 2 by 2 cube, anchoring it to the edge beads of this face.

I’ve made a new cube with different coloured beads to highlight the edges of the top face. Here’s how the first unit is attached:

beadmechanics_cube_step3a

First, make the bottom of the new CRAW unit as so:

beadmechanics_cube_step3bi

Then add the sides and top to complete the unit:

beadmechanics_cube_step3bii

We’re now going to work along the edge by adding a second CRAW unit next to the first in a similar way:

beadmechanics_cube_step3c

Now we’re at the corner. This corner CRAW unit is different, it has no edge bead to start from so is just joined to the previous (blue) unit on one side and is suspended in mid-air!

beadmechanics_cube_step3d

The next edge and corner is completed as before – just keep working around the next two sides:

beadmechanics_cube_step3e

Continue around onto the last edge, adding one corner unit and one edge unit.

The second edge unit on this last edge is going to be slightly different, since it has to include a bead from the very first unit we made:

beadmechanics_cube_step3f

Here I’ve added the bottom of the unit, the next RAW face is at the back and includes a bead from the previous unit, a new bead, a bead from the very first frame unit, and an orange edge bead from the central cube. The rest of the unit is then completed as before.

Finally, make the last corner (easier as you now have 2 units to join it to!):

beadmechanics_cube_step3g

Step 4

We’re now going to work on two of the three blank faces. Now is a good time to switch to a curved beading needle, as it will make stitching through the existing beads much easier. Start by turning the beadwork upside-down, so the completed frame is facing downwards:

beadmechanics_cube_step4a

Now start by adding a row of 4 CRAW units along one side, building off the existing beadwork as necessary:

beadmechanics_cube_step4b

Continue on around the corner, adding 3 more units:

beadmechanics_cube_step4c

Now we’re going to work another row back the way we came. First add 4 units above the previous row:

beadmechanics_cube_step4d

Then continue on round the corner, adding 3 more units to finish that row:

beadmechanics_cube_step4e

Now we have two more faces almost complete!

Step 5

Rotate the beadwork around and add 2 units along the remaining edge, separating these two faces into two frames:

beadmechanics_cube_step5

That’s two more faces almost complete!

Step 6

We just need to add the final blank face on the top to complete the cube. I found it easiest to start in the middle and work out. Start by adding 4 CRAW units in the middle on top, like this:

beadmechanics_cube_step6a

Once all 4 are done it will look like this:

beadmechanics_cube_step6b

Just the outer edge left to do! Work around the edges, adding a single row of units all around the outside:

beadmechanics_cube_step6c

Keep going until the top face is complete:

beadmechanics_cube_step6d

And that’s it for the CRAW! The cube should now have 3 blank faces that meet at a vertex, and 3 frame faces that meet at the opposite vertex:

beadmechanics_cube_step6e

The last thing to do is to add the crystals!

Step 7

We’re going to add a 4mm crystal to the centre of each frame face. With the cube orientated as shown in the left-hand photo below, we’re going to follow the thread path shown in the right-hand photo from ‘a’ to ‘b’ to add a crystal to the top face:

beadmechanics_cube_step7

This is shown in more detail in the photos below. First work through the beadwork to exit the bead shown, then add the crystal and continue:

beadmechanics_cube_step7a

Work around through the beadwork following the threadpath shown above and stitch back through the crystal, like this:

beadmechanics_cube_step7b

Now add a crystal to the other two remaining faces. Note that point ‘b’ in the threadpath of the top face is point ‘a’ for the next frame (as shown in the right-hand photo above). Once you’ve completed this second face, the thread will be in the correct place to start the frame on the left-hand side. You should end up with all three crystals pointing towards the top vertex, like this:

beadmechanics_cube_step7c

Don’t end the thread yet, you’ll need it to complete the hinges.

Up to this point all the cubes are identical, so just repeat the steps above to make all 8 in the same way. Then it’s on to joining them together!

Adding the hinges and joining them together

We’re now going to add some modified right angle weave (MRAW) hinges to each cube and join them together. I’ve followed the same sequence of joining them as I used for the model, so you can refer back to the photos at the start of this post while joining the beaded cubes together.

Cube 1

The hinge pattern for cube 1 looks like this:

beadmechanics_cube_1

The number next to each hinge is the cube it joins to. The first hinge we’re going to add is the one labelled ‘4’ in the photo above. Turn the cube so it’s orientated as shown in the photo below and work through the beadwork to exit from the bead shown:

beadmechanics_cube_cube1_a

Now we’re going to add the hinge beads along the edge using the MRAW thread path (see the beginning of the post for more about this stitch). I’ve used orange for the hinge beads so they show up clearly.

First add one bead and continue on through the top of the next MRAW unit (left photo). Then loop around the top of this unit, through the beads indicated in the right photo:

beadmechanics_cube_cube1_b

This extra loop makes sure the hinge beads are held in place nice and securely. Continue along the edge adding in two more hinge beads, following the thread path as shown:

beadmechanics_cube_cube1_c

You want to keep a fairly medium tension here – too loose and the hinge will be unstable, but too tight and you won’t be able to join it to the next cube.

Turn the beadwork so you’re looking at the hinge from the other side. Work along through the beads following the MRAW thread path as shown on this side:

beadmechanics_cube_cube1_d

The hinge beads should now be held in place with stitches on both sides.

Add in the second hinge (the one labelled ‘2’ in the photo) in the same way, so you have a cube that looks like this:

beadmechanics_cube_cube1_e

Weave the end of the thread into the beadwork so it’s secure, and trim it off.

Cubes 2 – 7

For cubes 2 to 7 you’ll add one new hinge, and join to another on a previous cube. For each cube I’ve marked on the photo which other ones they join to, and indicated which is the shared hinge with a circle around the number.

Cube 2

Here’s the hinge pattern for cube 2:

beadmechanics_cube_2

This cube joins to cubes 1 and 3, as labelled. I’ve put a circle around the shared hinge – you don’t need to add this one, just join to it.

So for cube 2 we’re first going to add the hinge labelled ‘3’ using MRAW, just as for cube 1. We’ll end up with a cube that looks like this:

beadmechanics_cube_cube2_a

I’ve made it a slightly different colour so you can tell which cube is which! Now we’re going to join it to the shared hinge on cube 1. Bring the two cubes together as shown on the left:

beadmechanics_cube_cube2_b

The edges we’re going to join are marked with a dot. On the right I’ve moved the cubes so these edges meet.

Join cube 2 to cube 1 following the exact same MRAW thread path as before – only this time you don’t need to add an extra bead, just use the one from the hinge on cube 1:

beadmechanics_cube_cube2_c

The MRAW thread path is exactly as before:

beadmechanics_cube_cube2_d

Once you’ve done one side, rotate the hinge so you can reinforce the other side:

beadmechanics_cube_cube2_e

That’s it, you’ve joined the first two cubes together!

beadmechanics_cube_cube2_f

Cube 3

The hinge pattern for cube 3 is shown below – this one joins to cube 2 using the existing hinge (marked with a circle).

beadmechanics_cube_3

Cube 4

Cube 4 joins to cube 1 on the hinge labelled below:

beadmechanics_cube_4

Cube 5

This cube joins to cube 3 as shown:

beadmechanics_cube_5

Cube 6

Cube 6 joins to cube 4:

beadmechanics_cube_6

Cube 7

This cube joins to cube 5:

beadmechanics_cube_7

Cube 8

The final cube is a bit different – you don’t need to add a new hinge, just join it to the existing hinges on both cube 6 and cube 7 as shown:

beadmechanics_cube_8

The finished cube

That’s it! You should now have a complete folding cube!

beadmechanics_cube_verrier

If you want to share some photos of your completed cube I’d really like to see them! You can always head over to my facebook page and post them there!

Beaded machines, Beadwork

Folding Cube

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:

cube2_verrier

and the other has crystals embedded in it:

cube1_verrier

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!

Beaded machines, Beadwork

Kaleidocycles

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!

kaleidocycle1_beadmechanics

And here’s the opposite set of faces:

kaleidocycle2_beadmechanics

Finally, here’s a view from the side:

kaleidocycle3_beadmechanics

Can’t wait to start the next one!