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!


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


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:


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


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


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:


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


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:


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


(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:


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:


That’s the second tetrahedron completed!


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:


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


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:


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:


Now pass the working end under and through the loop:


You should now have something that looks like like this:


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


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:


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


When joined it should look like this:


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:


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!