A few months ago, I went to a creative origami lunchtime session organised by some lovely people at `$WORK`

. I’d done origami a bit when I was younger, but mostly just frogs and cranes, which have since helped me while away the hours when invigilating exams. However, at this lunchtime session I was shown how to make modular origami. This involves making lots of (generally quite simple) origami parts, and then slotting them together to make larger structures.

I went home with a simple 12-unit Sonobe ball that afternoon and was very pleased with myself.

Things have rather escalated from there.

Between running some of these lunchtime sessions myself now, and being asked on several occasions on Twitter about how I make the pretty things I keep tweeting, I thought it’d be useful to put together a quick guide (or a link-farm, at least)..

Sonobe units are very easy to fold, quite forgiving, and can be used to make a cube (6 units), a cumulated octahedron (12 units), a cumulated icosahedron (30), and a kind-of truncated icosahedron (90, basically a spiky football). They’re a pretty good introduction to the general principles:

The 30-unit ball has the symmetries of an icosahedron (or dodecahedron). Once you’ve learnt how to construct that object in Sonobe modules, you’ve essentially learnt how to construct any 30-unit modular origami ball: they mostly involve slotting 30 edge-units into groups of three to form the 12 pentagonal faces of a dodecahedron (or equivalently/alternatively, slotting them into groups of five to form the 20 triangular faces of an icosahedron – the difference is mostly one of perspective). There are lots of variations on the Sonobe unit you can (re)invent, by adding back-folds that expose the other side of the paper, or that make the tabs narrower than the pockets, giving a more intricate look. Although the 90-unit structure is quite stable, the next one up (270 units) tends to sag under its own weight over time, but by that point it felt like a right of passage to make one. The Sonobe units can also be assembled inside-out to make inwardly cumulated polyhedra… …and they can also be assembled in pairs and then assembled into a spiked pentakis dodecahedron... …and other structures. The next unit I tried was the Penultimate edge unit (attributed to Robert Neal), which can be used to make a wireframe dodecahedron, as demonstrated by Matt Parker, the stand-up mathematician. Other variations of this subunit can be used to make pretty much any other wireframe polyhedron. Thomas Hull’s PhiZZ edge unit makes similar wireframe structures, but the modules fit together more tightly and the resulting structures are much more robust than you get with the penultimate modules. You can also make colour-change variants using the technique shown in Lewis Simon’s decoration boxes. For structures based on dodecahedra/icosahedra and made from edge-units, you can always get away with using just three colours and never have two of the same colour pieces touching. This is because you can draw a Hamiltonian circuit on a dodecahedron: that is a path from vertex to vertex that only visits each vertex once, and which comes back to where it started. You can represent this in 2D on a Schlegel diagram. If you colour alternate edges of the Hamiltonian circuit in two of your chosen colours, and the rest of the edges in the third, then you’ll avoid having any colour-clashes. I only learnt this after I started making these structures, so not all of them have this optimal colouring! The same 3-colour rule is true for the other Platonic solids, and also for the truncated icosahedron.Francesco Mancini’s star-holes kusudama uses a similar module to the PHiZZ, but with a little back-bend that gives a nice 3D star effect. This one is a dodecahedron-shaped (30 units), but a 90-unit truncated icosahedron should also be possible.

UPDATE: yes, it is possible 🙂 Lewis Simon and Bennett Arnstein’s triangle edge unit can be used to make very nice patchtwork tetrahedra, octahedra and icosahedra. They’re a bit fiddly to put together but are very robust once constructed. A similar patchwork effect for the dodecahedron can be achieved with M. Mukhopadhyay’s umbrella module; Sonobe units can be used to make analogous Battenberg-cake style cubes. The simple isosceles triangle unit (attributed variously to M. Mukhopadhyay, Jeannine Mosely and Roberto Morassi) can be used to make small and great stellated dodecahedra. The small stellated dodocahedron is particularly pleasing and makes a fairly robust decoration if made of foil-backed paper. The great stellated dodecahedron can be made from the same subunit, but is tricker to construct because a tab has to curl around into a pocket that is partly inside the next tab round. I used needle-nosed forceps to construct this, and I’m still not terribly happy with the result.The opposite is true for Paolo Bascetta’s star module, which makes a great great stellated dodecahedron, but a rather *eh* small stellation. This module needs duo paper (*i.e.* paper that is coloured on both sides) for best effect.

*and*squares (all of the polygons found in the Platonic solids)… …and also a pair of snub-cubes, which are even more interesting as the snub-cube has two non-superimposable mirror images, like hands, amino acids and amphetamines. I found Maria Sinayskaya Etna kusudama in Meenakshi Mukerji’s

*Exquisite Modular Origami*book. It’s a really pretty model, and robust once it’s assembled, but it can be a bit fally-aparty during construction: I used very small clothes pegs to hold it together as I was making it. Dennis Walker’s compound of five octahedra is also a bit fally-aparty, but I like it as – unlike many of these models – it genuinely is the polyhedron so-named, rather than something where you have to squint at the holes in the wire-frame and imagine faces there. The five intersecting tetrahedra are actually a lot easier to make than they look. Francis Ow’s 6-degree modules themselves are easy to fold, and the vertices are a lot more robust than they might appear. The most difficult bit is getting the modules interlinked in the right way. I’ve managed it twice, but only whilst staring at the YouTube video and performing assorted “purple = green” gymnastics in my head. Michał Kosmulski’s page has lots of lovely illustrations, instructions and inspirations. I found Tung Ken Lam’s blintz icosadodecahedron (also credited as Francesco Mancini’s UVWXYZ intersecting planes model) there. It has the same symmetry as the Electra icosadodecahedron above, but you can see the six intersecting pentagons more clearly. Both have the same underlying structure as the Hoberman sphere – that expanding/contracting plastic stick model thing beloved of science fairs. This last one is a bit of a cheat as (in theory, and mostly in practice too) the structures above are held together by nothing more than friction. Valentina Gonchar’s revealed flower star kusudama has to be glued, which is kind-of cheating, but I couldn’t resist as it is two structures in one: Things I’d still like to do:

- Build a much larger PhiZZ ball (270 units): this would be useful for demonstrating the structures of viral capsids. UPDATE: Done!

- I’ve not yet found a good great dodecahedron model: they exist on Pintrest, but I’ve yet to find any instructions for one.
- I have lost wherever it was I found the instructions for this inwardly cumulated rhombic triacontahedron: I’d quite like to rediscover them so I can credit the inventor! UPDATE: this isn’t where I originally saw it, but AresMares by Gewre has a video tutorial.

- Invent my own module 🙂

## 15 comments

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Amazing and inspiring! You have blown my mind. I’m going to go make some phizz units with the stripey color changes. I wish I’d thought of that!

Thanks for sharing all this neat stuff, it’s given me a lot to try out. If you’re curious, I have a module that’s morphologically similar to the Sonobe unit, that I was taught by a Japanese woman who went to my church when I was a child. It’s not a huge revelation but it creates an interesting look if you’re looking for something new to mess with.

Author

I _am_ curious – do you have a folding diagram?

These are so beautiful, thanks for sharing! Unfortunately, I have been tearing my hair out trying to figure out how to make a few of these designs, specifically the triangular tetrahedrons and the rhomboid/kite-like octahedron. I’ve searched the internet and I’ve been to the library to check out several origami books, but I cannot seem to find a way to make them! I’ve even tried to deconstruct them on paper to try and see how the units fit together, but that is a mess best left forgotten.

In any case, here’s a url that has screenshots of the models that I’m having trouble with, if you could help clarify how to make them I would be most appreciative!

https://imgur.com/a/Yv1WsLo

The first one I designated the kite-like octahedron, the second the simple triangular tetrahedron, and the third the complex triangular tetrahedron.

Author

They’re both made using the same triangle edge module as is used to make the patchwork icosahedron in the same image. When you make the icosahedron, you arrange the 30 modules in groups of fives around each vertex, but for the octahedron, you arrange 12 modules in groups of fours around each vertex – same technique as for an octahedron made of sonobe modules. The tetrahedron requires just 6 triangle edge modules, arranged in threes – it’s quite difficult to get all the bits tucked in, but not impossible.

Author

The very simple tetrahedron I think I made from just two modules tucked into each other, but I can’t remember off-hand which kind. It may just have been two sonobes, possibly one left-handed, and the other right-handed? I’ll take a look tomorrow if I have time.

I have recently come up with a variant on the sonobe model that gives a module easier and cleaner to assemble for the inverted icosahedron build, and it keeps all the versatility of the regular sonobe . I don’t know if its original though. I could send you the instructions if you want

(I still call BS on the bascetta star. He supposedly created the model in 2007, but the model was taught to me in spring 2005…)

Author

Apologies for the extremely late reply – I would be interested in the sonobe variant if you’re still prepared to share?

I AM SO HAPPY! I FINALLY COMPLETED THE INWARDLY CUMULATED POLYHEDRA!

Author

🙂

Your work is really amazing!

I’m trying to work my way through the Sonobe-Family from 6 to 270 with 3 different colours. The 6, 12 and 30 are possible with 3 colours. But I’m not sure if the 90 and 270 are also doable with 3 colours. I have built the 9, but it’s not working for me with 3 colours.

I would like to know if you could tell me if it is even possible to make a 90 and 270 Sonobe-Ball with 3 different colours. And is there a formula or something like that to answer my question? And if it is possible, is there a trick in assembling it?

Greetings

KL

Author

You can definitely use just 3 colours for the 30 (‘icosahedron’) and 90 (‘truncated icosahedron’) using the Hamiltonian circuit diagrams – there’s a link to the 90 unit version in the post above. I don’t know whether it is possible for the 270. It’s much easier to actually make a perfect 3-colour model with the PhiZZ units than with the Sonobe – the Schlegel diagrams shown in the post above re much easier to interpret because PhiZZ gives you a wire-frame look that’s easy to map onto the Schlegel diagram. With Sonobe, you have to try much harder to visualise where the edges actually are. It’s definitely do-able, but I would suggest trying a 3-colour 90 PhiZZ first to help get your eye in.

Is there a way to make a ball with more than 270 units?

I have already built (and glued) the 270 unit stucture.

But, I was wondering if instead of the 5-6-6-5 peak formation, will a 5-6-6-6-5 formation work as well?

And if so, how many pieces are required?

Yes, this should be possible, have a look at Tom Hull’s video explaining how to make larger Buckyballs using his PHiZZ unit: https://youtu.be/9GmSkfBt8fo (although this applies generally and not just to using his PHiZZ unit).

Great work Polypompholyx, you certainly seem to be enjoying yourself exploring the world of geometry through origami 🙂

I think the Compound of Five Octahedra is actually a design by Meenakshi Murkerji, assuming it is 30 and not 60 pieces?

Also, your last model is a design by Silvana Betti Mamino and I believe is often called Silvana’s Star Ball. HTH 🙂