In February of 2025, Tom Montenegro-Johnson, professor of applied maths at the University of Warwick, approached us with a proposal—to spend three full days collaborating with paper artist Coco Sato on her next piece. Tom and Coco have worked together for years, developing inspiring pieces motivated by the fluid dynamics that Tom uses in his cutting-edge research, including an evocative public engagement piece for the University of Birmingham which saw hundreds of students create origami butterflies. Now, we had to leave our algebra in the office and spend some time reflecting on how our research could be made into artwork. Even more dauntingly, we then had to make it happen. Fortunately for us, origami and fluid mechanics are more closely related than one might expect at first glance.
Taking shape
Origami, the art of folding paper to form complex geometric structures, has been practised in Japan for centuries, with references to paper cranes and origami butterflies dating as far back as the 17th century. Interest in origami was originally purely for recreational and decorative purposes, but over the years origami structures grew in complexity and nowadays, people are looking to origami not only as a fun creative outlet but for new innovations in science and engineering.
Techniques from origami can be used to build shape-transforming structures. An example is the folding mirrors used in the design of the James Webb Space Telescope: these mirrors can fold into compact formations suitable for rocket travel, and unfold after launch into the convex or concave shapes required for distant infrared light collection. A video of the telescope unfolding is available on YouTube, and on Nasa’s website there are even instructions to create your own James Webb origami. The same ideas can be used to create medical devices, such as a stent (a device used to hold open a vessel in the body which has lost structural stability). Origami engineering has allowed new stents to be built which can be delivered with minimal invasion in their compact ‘folded’ state, and then unfold only once perfectly in place.
Our research group specialises in the mathematical modelling of hydrogels: these are soft, hydrophilic gels made from networks of polymers which can swell to many times their original volume through absorption of water. Some day-to-day hydrogels you may have encountered include soft contact lenses, blister plasters, and the children’s toy Orbeez. Hydrogels are useful for a wide variety of medical applications because their ability to hold large amounts of water makes them highly biocompatible (meaning that they have similar material properties to human tissue).
Orbeez are made of hydrogel
They are also useful as actuators, because they can be designed to respond to changes in their environment, and hence exert forces and torques on nearby objects. For example, a hydrogel could be programmed to swell in response to increased temperature, which could then trigger the folding or unfolding of a larger structure. Hydrogels which respond to their environment in this way are called smart hydrogels. Below are two classic designs for hydrogel actuators:
In the first example, known as a bilayer, the upper hydrogel layer expands, causing its length to increase, which then forces bending in the lower layer. The second example, the box hinge, has a similar mechanism, but produces a sharper fold. With these ideas, hydrogels could be used as a trigger to fold and unfold an origami-inspired device.
These are very simple actuator designs, but by programming hydrogels to respond to external stimuli precisely in a desired way, they can be used to build soft robots, and even ‘hydrogel brains’—a recent study designed a hydrogel which could play Pong!
We brought together these ideas of origami engineering, and programmable, smart hydrogels to create the artwork you now see on the cover of your copy of Chalkdust.
Creating origami
We were inspired by some of Coco’s previous works with modularity for this piece. In particular, we loved the almost algorithmic nature of some of her tessellating pieces, where lots of smaller origami structures come together to display a bigger picture—much like how it takes many pixels to create an image on a screen.
The individual units that we need to fold for this piece are called ‘Sonobe units’. Though the origin of the name is unknown it is shared with a town in Kyoto, Japan. These units are commonly used in mathematical modular origami as they are relatively beginner friendly to build, and when combined in 6, 12 or 30 units can make a cube, octahedron or icosahedron respectively. See ‘How to make a Sonobe unit’ in the magazine to make your own!
These units are then combined together in a repeating pattern as laid out in the below photo of Coco’s previous artwork.
Sonobe units. Image: cocosato.co.uk
In our case, we wanted to showcase our research on responsive hydrogels, so we pieced our Sonobe units together to make a picture of a swelling hydrogel tube, inspired by recent work by Tom and our colleague Joe Webber:
A simulation of a responsive hydrogel pump from a recent paper by Joe Webber and Tom Montenegro-Johnson, which inspired our origami art.
Heating the tube at the left end causes the gel to start shrinking and expel fluid, creating a pump mechanism. The swelling of the gel is governed by the advection–diffusion equation
\begin{equation*} \partial_t \phi + \boldsymbol{q} \cdot \boldsymbol\nabla \phi = \boldsymbol\nabla\cdot[D(\phi)\boldsymbol\nabla \phi]. \end{equation*} Here, $\phi$ is the porosity of the gel—at each point in the gel, the value of $\phi$ conveys what fraction of the gel is water and what fraction is solid matrix. A porosity of 1 therefore corresponds to pure water, while zero corresponds to completely dry. The vector $\boldsymbol{q}$ represents the overall velocity of material, accounting for both the water and solid matrix, and $D(\phi)$ is a diffusion coefficient determined by the chemical properties of the gel.
Encoding equations
We wanted to link our hidden message back to the exciting prospect of ‘soft computers’ by ‘programming’ a secret message into our origami. There are many ways to hide a message. The entire field of cryptography was generated for such a purpose. We decided to find a way to convert the governing equation for hydrogel swelling into a binary string. It becomes relatively simple to convert between different number systems, but our equation has more symbols than numbers within it so we needed to translate first into a number system.
Given that mathematicians will use an encoding language to type up maths equations onto a typesetter, we chose to begin from the LaTeX of the equation:
\partial_t \phi + q \cdot \nabla \phi = \nabla \cdot [D(\phi)\nabla \phi]
We need to convert this to a string of ones and zeros, but first converting it into a string of any numbers will do. Luckily in 1963 an IBM engineer faced a similar challenge, and the American Standard Code for Information Interchange (Ascii) was devised as a character encoder. Ascii assigns a unique numerical code, ranging from 0 to 127 to each upper case letter, lower case letter, digit and symbol. Converting this text string of LaTeX code will then give us a series of 66 numbers—one step closer to our goal!
Finally, we convert the string of numbers into its binary representation. When we write numbers in decimal, each new digit represents how many (between 0 and 9) of that power of 10 we have, so right to left we have units, tens, hundreds, thousands, etc. In binary we do something similar. So the first entry is twos, then fours, then eights, then sixteens, and so on. If we want to do the conversion by hand, an easy method is to continually divide the number by two, at each step document the remainder (either 1 or 0), and then read the number backwards. For example:
\begin{alignat*}{2} 53/2&=26 &\text{ remainder }1,\\ 26/2&=13 &\text{ remainder }0,\\ 13/2 &= 6 &\text{ remainder }1,\\ 6/2 &= 3 &\text{ remainder }0,\\ 3/2 &= 1 &\text{ remainder }1,\\ 1/2 &= 0 &\text{ remainder }1. \end{alignat*}
So, 53 becomes 110101.
We want to encode each of the numbers to have the same number of digits, so we’re going to commit a little bit of bad maths and stick some zeros at the start of the numbers so each number is the same length in binary. Our biggest possible number is 127 (the largest Ascii value) which is 01111111, as 128 is the 8th power of 2. With this we made every number in our array 8 bits long. This means for our string of 66 numbers, we would have $66\times8 = 528$ ones and zeros to encode in our artwork in order to capture the full equation. In the final piece, lighter colours represent zeros and darker colours represent ones.
Put it together and what have you got?
We had three days to bring this work to life, and after day one the six of us (four postdocs, Tom and Coco) were only able to fold approximately 300 modules, it became quickly apparent that reinforcements needed to be called in. Luckily the PhD students at the University of Warwick delivered; with many hands on deck we were able to complete folding all of the individual units by the middle of day three (after a quick break for an incredible guest seminar by Emma Bouckley, University of Cambridge, as part of the soft matter seminar series). Then we were back at it to combine the units together and attach it to the boards.
Overall there was something incredibly satisfying about combining maths and art, and Coco was a fantastic guide into the world of paper artistry. We were able to bring to life a new method of data visualisation, and enjoy all the messiness and chaos that came with it. As mathematicians, many of us felt like creating art was not for us, but this work made us see that art can take many forms, and that we all have unique perspectives to contribute. Perhaps we are more artistic than we think.
The final piece, by Coco Sato, Tom Montenegro-Johnson, Daniel Booth,
Emma Bouckley, Ellen Jolley, Kat Phillips, Joe Webber and volunteers from the University of Warwick.
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