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!