Beadwork, Polyhedra

Five Intersecting Tetrahedra

Earlier this year Diane Fitzgerald posted a geometric challenge: make a beadwork version of the origami model of five intersecting tetrahedra. After many months of work here’s my version!


I’m not the first person to make this – Kris Empting-Obenland gets the credit for that! Her beautiful version has been accepted to this year’s Bridges conference – you can see it in the Bridges 2019 gallery here!

Five intersecting tetrahedra is an interesting compound shape that has some similarities to a dodecahedron. I’m always interested in geometric challenges so I decided to see what I could do. My first attempt was with 30mm bugles, which worked surprisingly well!


If you want to try making one of these I recommend working from a video showing the construction of the origami model, like the one here. However, I definitely recommend using similar colours to those shown in the video – and not what I did, which was mostly similar colours but in a different order which got very confusing!

My next step was to take a few ideas I had been working on following on from my Sunburst dodecahedron, which uses Sue Harle’s tubular diagonal peyote technique.  I realised I could use this technique here to make individual beams for the sides of the tetrahedra, meaning the construction would be very similar to the origami version – and hopefully that it would be easy(ish) to join together. Here’s my first attempt at a tetrahedron using this method:


It worked! My next step was to work out the dimensions of the beams so that the compound would work – if the tetrahedra are too small then you can’t interlink them, if they’re too big then it won’t hold its shape. Because the geometry of the beams here is very slightly different to the origami version they need to be a very slightly different size so the model will fit together correctly. So there was a brief interlude from the beadwork for some maths to work out the exact size, which turns out to be a ratio of width to mid-beam length of 1:13.5441. I checked this calculation three times and then asked someone else to check it, as I had nightmares of making 5 tetrahedra that were all slightly too small to fit together!

The next step was to start beading. I spent a lot of time measuring and re-measuring the first few beams to check the size (I’m very glad I invested in some digital calipers a while ago!) but eventually managed to make a full size tetrahedron – 1 down, 4 to go!


I decided to use just one colour, which also gave me an excuse to use some Crystal Marea delicas that I’d been wanting to make something with for ages!

The next step was to make the second tetrahedron. This one was easy to assemble around the first, and it seemed like I was making reasonable progress.


However, experience from the bugle bead version (or rather, experience unpicking many mistakes in the bugle bead version) left me feeling a bit apprehensive when it came to the next tetrahedron – and I ended up leaving the project to languish for a few months.

To try and make it easier to pick up I tried making another bugle bead version – this time all in the same colour. However, this ended up being so difficult it didn’t really help my confidence at all! I decided to just make all the beams for the remaining three tetrahedra then set aside a long weekend to try putting them together. Here are all the pieces ready for the final assembly:


I was dreading this bit, so decided to just tack the beams together at the ends with separate thread to make it easy to take apart if (when) I made a mistake. To my surprise though it went together fine – I’d like to think it was all the practice with the bugle bead versions, but I think it might just have been luck! Here’s the initial assembly, before I stitched the beams together properly – it’s a bit of a tangle!


But it gradually sorted itself out into something more symmetric and geometric!


At this point it became clear that the beams were the right size – which was a huge relief! Then it was just a matter of stitching in the last few threads to finish the piece and complete the challenge!


And this definitely was a challenge! But I’m glad I attempted it as it pushed me to develop a few new construction ideas, and even though at times failure felt inevitable I did enjoy the process! Also, at about 65g of delicas this is easily the largest piece I’ve ever made!

And as for that second bugle bead version? I did eventually manage to get it to work – and it’s now one of my favourite pieces!


Beadwork, Polyhedra, Tutorials

Truncated Octahedron Hyparhedron

Here are some brief instructions for the truncated octahedron hyparhedron. This is actually a pretty simple shape to make. It’s just 4-hats joined together with a few extra warped squares. If you know how to zip together warped squares to make a star you can use the same method here! My warped squares are 7 rows in total, and I make them out to row 6 then use row 7 to zip to any other squares as needed.


First join four warped squares with 2 brown sides and 2 green sides into an upside-down 4-hat – that is, with all the points in the centre pointing downwards. This will be one of the square faces you can see in the photo above.


Here’s a diagram for the individual warped squares that make up the 4-hat:


Make the first square completely all the way out to row 7 and stitch in the threads (the photo is in red and white instead of brown and green – sorry!):


Now make a second square out to row 6 and join it to the first square on one brown (or red!) side as part of row 7 as shown (note I’m working anticlockwise around the square):


Finish the round and weave in the end – you should now have two squares joined together like this:


Make a third square out to row 6 and again join to one of the others on one brown (/red) side as part of row 7 as shown:


When this square is completed it will look like this:


Make a fourth square out to row 6 and this time join it to the two remaining brown (/red) edges from the previous squares using row 7:


When this square is complete you will have a finished 4-hat like the one below!


Here it is from the side – the centre points downwards (so technically it’s an upside-down 4-hat!):


Make five more of these so that you have six identical 4-hats in total. The warped squares here are all edges of a square face on the finished shape.

Now make a completely green warped square out to row 6 (I use the same silver diamond pattern as before, but all the sides just have the same background colour). Step up for row 7 and zip it on all sides to two of the 4-hats, as shown on the left of the diagram below. The centre pyramid of both 4-hats should be pointing downwards. (Note that I’ve shown this new warped square in blue rather than green!) The new warped square is an edge of a hexagonal face. To show the shape flat I’m going to draw the warped squares slightly distorted (as on the right of the diagram) from this point onwards.


Here are two 4-hats and a warped square ready to be joined together:


Here are the first two sides being joined together:


And here’s the piece from the other side showing last two sides being joined together:


When the join is complete the beadwork will look like this:


Here’s another in-progress photo from slightly later in the construction outlining how this square fits between two of the upside-down 4-hats:


(Note though that this particular photo used a slightly different order for joining the squares than the instructions here!)

Repeat this step three more times to join three more 4-hats around the first, as shown in the diagram below:


Now join in four more warped squares around the edge of the shape connecting some of the remaining edges of the 4-hats as shown below:


The diagram above looks pretty distorted but in reality the warped squares will fit easily into place.

Turn the beadwork over. There will be a space for the remaining 4-hat, which should be joined in using 4 more green warped squares, as shown below:


Once all these joins are finished the hyparhedron is complete. Sorry the instructions are a bit brief but if you have any questions just ask and I will try and help!



© Copyright 2019 Patricia Verrier. All rights reserved.

These instructions are for personal use only. Please contact me if you require more information.

Beadwork, Polyhedra


Hyparhedra are polyhedra made from hypars – hyparbolic paraboloids – a shape more commonly known in the bead world as a warped square. I was introduced to the idea of hypars and the academic work on them a while ago by the Contemporary Geometric Beadwork project.

Polyhedra made from hypars were named hyparhedra by Erik Demaine, who wrote a paper (“Polyhedral Sculptures with Hyperbolic Paraboloids”, Demaine, Demaine & Lubiw, 1999) on a method of making any polyhedra from paper folded hypars. There are details of the method and photos of the shapes here. It’s based on making “k-hats” – pyramids made from hypars – and using them as the faces of polyhedra. They demonstrated how to make a variety of shapes including all the Platonic Solids – that’s the five possible polyhedra made entirely out of the same regular shape: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.

You may also notice on that page that one of the first shapes is recognisable to the beader as a warped square star! (In their paper they mention that the earliest example of a warped square star in origami is from 1958!) This method also works with beadwork hypars – Joke van Biesen and Kim Heinlein have both made beautiful beadwork versions of the hyparhedra cube from the paper.

The photo below shows a beadwork 4hat – four warped squares joined together on two sides each to make a 4-sided pyramid. Each of the warped squares is half cream and half red, so that the pyramid in the centre ends up one colour and the edges another. I’ve outlined the warped squares in green on the right to highlight how they’re joined together.

k_hat_BeadMechanics k_hat2_BeadMechanics

I’ve been playing around with a slight variation of the idea for some time now. Instead of the method using k-hats as the faces, I fold one warped square over each edge of a polyhedron and join them so that their points are at the vertices and centres of each face. This method seems to be fairly well known in the origami world, but I haven’t found anything about it in the mathematical literature so far. I have however seen numerous independent discoveries of a dodecahedron made this way in the beadwork world – the infamous rhombic hexacontahedron! I think the dodecahedron is the only Platonic solid that will work this way, so I decided to try using this method to make various Archimedean solids.

So far I’ve completed a truncated tetrahedron, the dual of a cuboctahedron (a rhombic dodecahedron) and a truncated octahedron:


I initially didn’t think that these would work as their dihedral angles are quite small (that is, they have “sharper” edges compared to larger shapes), but the beadwork turned out to be more flexible than I thought! Here’s a photo of the truncated tetrahedron from a different angle showing how much some of the squares will distort:


Here’s an more detailed illustration of the method using the truncated octahedron – a shape with 6 square faces and 6 hexagonal faces. The photo below shows the outline of the truncated octahedron over the hyparhedra:


And this photo shows how a warped square has been placed over an edge with the two “upwards” points at the corners and the two “downwards” points at the centre of the faces:


Interestingly the shape formed by this hyparhedra is quite similar to the third stellation of the rhombic dodecahedron. Which brings me back to the rhombic dodecahedron hyparhedra:


This shape was interesting – trying to put warped squares over the edges of a cuboctahedron didn’t work as the surface ends up curving in the wrong direction. The squares want to be the opposite way around – so that their upwards points are in the centre of the faces not at the vertices. This just creates the “dual” polyhedron – the shape you get if you join the centres of the faces of a polyhedron together. In this case it’s a very pointy rhombic dodecahedron!

I’ve started a snub cube which seems to be working out the same way and looks like it will end up as it’s dual shape too. I have a rhombicuboctahedron in progress too but I’m not sure if it will work as itself or as it’s dual yet.

I originally started all this with a truncated icosahedron. However I couldn’t decide on colours so ended up with a lot of individual pieces and not much completed. This is the main reason I’ve switched to using the picasso delicas for all these shapes – it gives me a few colours options without being overwhelming. So I’m restarting the truncated icosahedron in red and brown to match the two other truncated shapes I’ve made so far!

The biggest piece I have in progress though is a rhombicosidodecahedron, which is working well but is also going to be very big! Here is the piece I’ve completed so far (about a sixth of the total) compared to the other shapes:


There’s only one other Archimedean solid larger than this – a snub dodecahedron which needs 150 warped squares! I’m wondering if this one will work as its dual since this has happened with some of the other shapes with a lot of triangular faces.

There are also some Archimedean solids with 8 or 10 sided faces, which will not be flat when made this way with warped squares (since warped squares make flat hexagons when joined on two sides). However, I’ve tried joining 8 warped squares together as the start of a truncated cube and I think it will still work!

The different shapes and surface curvatures that can be generated by this hyparhedra method are interesting – part of my reason for working through all of these systematically is to gain a better understanding and feeling for how to make more general surfaces so I can create more complex shapes. I’ve already used the ideas in several other shapes – for example the rick rack dodecahedron and variation are joined together with warped squares making up the edges of the underlying polyhedra. It was the hyparhedra idea here that helped me work out how make these two pieces!

If you’d like to try making a hyparhedra here are some brief instructions on how to make the truncated octahedron!

Bangles, Beaded machines, Tutorials

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!


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Polyhedra, Tutorials

Truncated Tetrahedron

When I made the Sunburst dodecahedron I thought that the technique could be easily adapted to make other polyhedra. The flexible nature of the edges make it easy to adapt to shapes with different angles between the faces. I recently tested this idea by making a truncated tetrahedron, the piece below is the result:


A truncated tetrahedron has four hexagonal faces and four triangular faces, so the resulting shape looks quite complicated, and looks very different from different angles!


It’s a bit smaller than the original dodecahedron, but not by much. Here they are side-by-side for comparison:


I really enjoyed making this piece and I was pleased by how easily the components could be used to make both triangular and hexagonal faces. I have a lot more ideas for other shapes now too!

Instructions for both these pieces are in the Sunburst tutorial in my Etsy shop! (And a huge thank you to Sue Harle for permission to use her original diagonal tubular peyote technique in the tutorial!)

Beadwork, Tutorials

Warning about Phishing Websites

Please be aware that a number of phishing websites have come to light over recent days that have copied a large number of beadwork and craft listings from Etsy, apparently in order to scam people out of payment information. Unfortunately some of my tutorial listings appear on some of these sites. These are not genuine listings!

My tutorials are only available from my Etsy shop, Interweave and here on my website.

You can find a full list of those available on the tutorials page.

Finally, please be careful when following links from sites such as Pinterest – always check that the link is genuine before clicking! Be(ad) safe out there!

Beadwork, Polyhedra

Rick-Rack Polyhedra Variations

While I was making the first rick-rack dodecahedron I had an idea for a slight variation made by joining the rick racks together point-to-point instead of edge to edge. Since this would require a join between three edges, I first thought that I could use a warped hexagon instead of a warped square. However, I was completely wrong about that! The angle of the warped hexagon was far too small. After a bit of trial and error I found that three warped squares joined into a pyramid on two of their sides resulted in a triangular shape with the right surface angle. I then spent the better part of a day trying out different colour combinations and patterns to end up at the conclusion that they looked best just all in the dark blue colour. It’s a bit of extra work making the three squares for each join but they look great – the small pyramids create this extra spikey structure around the rick racks that I really like. Here’s the finished piece:


If you compare it to the original you can see the differences – in the new shape two rick-rack points meet at each join where in the original three points meet at each join:


You can see the difference in how the two are constructed in the diagram below – on the left is the variation and on the right is the original:

The two polyhedra – the original and the variation – correspond to a dodecadodecahedron (no that’s not a typo!) and a ditrigonal dodecadodecahedron. (Although the triangular faces on the new shape are convex not concave, but close enough!). The original shape is also similar to a hyperbolic dodecahedron (which I believe is a dodecahedron that tessellates in hyperbolic space, rather than is entirely hyperbolic itself).

I was also asked if it was possible to make smaller shapes such as tetrahedra or cubes out of rick-racks and warped squares. The crucial factor here is the angle of the warped squares – their maximum angle (the angle between the ‘V’ shape each one forms) is about the same as the angle needed to join the rick-racks together on a dodecahedron. They’ll squash down enough to join together rick-racks at a smaller angle, which should make shapes with higher dihedral angles (i.e. less pointy edges) possible, such as truncated icosahedra, but not those with small dihedral angles such as tetrahedra. Fortunately there’s an easy solution! You can make something something very similar to a warped square by joining two triangles together one on side, as shown on the below on the left, compared to the original warped square join on the right:

Unlike the square however, the triangles will bend to any angle along this join, which should make any regular or semi-regular polyhedra possible! To test the idea I made a tetrahedron using this method:


The rick-racks here have three peaks instead of five, but otherwise it’s made in exactly the same way as the original dodecahedron, just joining each one together with the pairs of triangles instead of with a warped square. The resultant shape is very pointy (and hard to photograph!) but much quicker to make!

I’ve now made a set of podcasts with various different numbers of sides so I can make a whole set of these polyhedra!