# Hair today: Ray Goldstein on ponytails and statistical physics

*Professor Ray Goldstein is Schlumberger Professor of Complex Physical Systems at the University of Cambridge, a Wellcome Trust Senior Investigator and was last month made a Fellow of the Royal Society. He also has an IgNobel award for the physics of ponytails. I asked him what a good bob has to do with statistical physics and his work on cilia.*

**Hello Ray. To start with, could you explain to me what “statistical physics” is exactly?**

Statistical physics is that branch of science that makes the connection between the way elementary objects like atoms and molecules interact at a microscopic level and the macroscopic observables of a system, whether it’s a fluid or a solid or something in between.

If you think about the number of molecules of water in a coffee cup, it’s unimaginably large. We’re not going to try to write down the dynamical laws for every single one of those molecules and somehow solve them, so we need to understand how to average over large numbers of individual elements. That as a scientific method was developed over the last century or so, and it’s been applied incredibly successfully to solids, liquids, gases, gels, in astrophysics, nuclear physics – essentially everywhere.

I got involved in the field as a graduate student, working on some problems involving transitions between states of matter. But thereafter I tended to move into problems involving dynamical phenomena and pattern formation in nature, such as patterns formed by chemically reacting systems or interfaces between different fluids. In biology, organisms such as slime moulds secrete and respond to chemicals, and these form beautiful patterns of chemical waves.

**So how did you end up working on ponytails?**

Out of the blue, I got an email from a colleague of Patrick Warren, a scientist at Unilever, saying in effect, “We know that you have some expertise in the study of elastic objects and filamentary structures in physics and biology. You might find it interesting to work with us on this problem involving the statistical physics of hair” – that’s how they phrased it. I thought this was a prank at first as I’d never thought of hair and statistical physics in the same sentence!

But when you think about it we all have something in the range of 50-100,000 head hairs. That’s a very large number, so even if I gather a small fraction into a bundle, there still are about 10,000 hairs. Hence, understanding the properties of a bundle of hair really is a problem of statistical physics. You’re dealing with large numbers of objects but they’re each unique. Most hair has a certain curliness to it on one level or another – it could be a gentle long wavelength meander or a tight curl, but every single hair will be slightly different.

Put a bunch of these together and the individual hairs collide with each other. It takes energy to compress the bundle because you are straightening out the curls and that is like a spring. So a ponytail is like a bunch of random springs, essentially, that have some general features in common but which are each fundamentally distinct.

In the first meeting we had with Unilever we were given these so-called ‘switches’ of hair. These are commercially fabricated collections of real human hair taken from donors all over the world that are used by the personal care product industry as a test of shampoos, conditioners and other hair treatments. So if you want to test Brazillian or Chinese or Canadian hair, you can order these. They have about 10,000 hairs and are clamped at the top with a metal clamp that’s glued to the hair. And when you hold it, it looks just like a ponytail.

I looked at this and thought, if there’s any chance of being able to come up with a mathematical description of what a bundle of hair looks like, why not start with a ponytail? Because really there are three things that are generically important to bundles of hair and you can see the effect of all of them in a ponytail. One is the elasticity of hair: it has a certain average shape and it resists bending around that average shape. The second is it has mass, so gravity is trying to pull it down. That’s opposed by the third thing, which is this random curvature that is somehow fluffing up, creating volume in the hair bundle. That’s when we finally set the task of explaining the shape of a ponytail.

**Why were Unilever interested in this?**

It turns out that one of the key questions for them is to understand hair tangling –something we think we can tackle now, following this work. People with long hair know very well that you comb, comb, comb and always end up with these tangles. One of the primary purposes of hair conditioner is as a detangling agent – it lubricates the hairs and makes it easier to comb through the tangles.

So Unilever thought it would be good to try and understand the physics of tangling: what makes some hair more tangle prone than others? How does one try to mitigate against tangling? What’s going on? It really is a problem of physics because these elastic filaments are colliding with each other and getting oriented by a comb and somehow they tangle. That’s the underlying question. The secondary question is what is the physics of the body or volume of hair that people seek out in a shampoo or conditioner.

**And so you approached it looking at each individual hair and its own characteristic physical properties.**

We first wanted to understand what the characteristics of individual hairs were, yes. The first task was to devise a nice imaging system where we could look at a single hair held under gravity or lying on a surface and determine its 3D shape accurately. This is not a trivial thing to do because hair is so fine – a single hair is a tenth of a millimetre across and maybe 25mm long, so to take a picture of it and somehow work out what its 3D structure is requires a bit of work.

We developed an algorithm and imaging method to do this. We could then start taking large numbers of hairs and get statistics on these curvatures. Then we started taking pictures of the ponytail switches, trying to characterise what the shape of a ponytail is, what is that bell-shaped envelope of the outside of the ponytail, and again that takes a bit of work. We developed a way to take the measured shapes of ponytails and extract information about the random curvatures and their effect on the bundle. We found a way to take the equation and use it along with images of real ponytails to figure out a simple description of the physics of these random curvatures.

**Which brings us to the ‘Rapunzel number’. What is this?**

If you were to take a hair from your head and cut it to about 2 inches long and hold it between your thumb and index finger, you would discover that independent of whether you held it so you pointed it down, up or to the side, it would remain straight. That’s because, at that length, gravity doesn’t have enough force to bend the hair. It’s why, if you have a short haircut, your hair sticks up straight – gravity doesn’t affect it.

It turns out that that length can be calculated as a balance between the stiffness of the hair – its resistance to bending – and its mass per unit length. So if you know the stiffness of the hair, you can calculate the length below which hair is not bent by gravity and above which gravity starts to have enough of a pull to bend it. And that turns out to be about 2 inches.

So the Rapunzel number is simply the ratio of the actual length of the hair to this 2-inch length scale. If the Rapunzel number is 1 the hair has a length of 2 inches. If the number is 5 it has a length of 10 inches. So when the Rapunzel number is small – 1 or less –gravity doesn’t matter. Where the Rapunzel number is large – 1 or more – gravity matters a lot. And it’s in the latter that you get the long, slender hanging shape of a ponytail. When the number is 1 or less you get something more like a paintbrush: a flaring, splaying shape of the hair, which does not depend on the orientation of gravity.

**What are some of the real life implications of this research?**

There’s been talk about loft insulation. In this case, it’s the Rapunzel number that is important because of the question of how easy it is to compress the bundle. Imagine you have a bunch of fibreglass fibres. If you push on it there’s a bit of a resistance to bending and the natural question is what is the relationship between the force you have to apply and the amount by which you compress it, the elasticity of the bundle.

That’s what the other part of the calculation we dealt with covers: the role of the curvature of the filaments and their interaction with each other. We derived something that is technically known as the ‘equation of state of hair’. ‘Equation of state’ is a term used to describe a gas – how much pressure does it take to shrink it into a certain volume. In our case, the equation describes the pressure necessary to compress a bundle by a certain amount. That’s the result that would probably have an even broader application.

One place where physicists have already made contributions involving hair is in the computer graphics industry. Next time you watch an animated film you’ll see that representing hair accurately is important and a challenge and you can immediately tell whether something is animated or not when it’s not done right. People have now gotten to the point where they can replicate the walking of an animal or human in a very realistic way. And now also the animation of hair motion is getting very realistic. But it has been a great challenge so it’s possible that some of the ideas we’ve put forward would help with more accurate computer animation of hair and other things.

**Were you surprised by the range of applications of this research?**

To be honest, no. The way the problem was presented to me, even at that early stage, it was very clear that hair and fur and fibrous materials are everywhere in nature. If you go back to the work of Leonardo Da Vinci, you will find that he was fascinated by the flowing hair that he saw – he drew a famous picture in one of his notebooks of turbulent water, going past the supports of a bridge where you see the kind of braiding flows that look a lot like beautiful long hair. He remarked that there was this visual connection between flowing water and hair. This was somewhere in the back of my mind and my collaborators’ minds; as we discovered the mathematics, we ended up using mathematics from fluid mechanics.

This problem of a bunch of long filamentary objects with random meanderings also shows up in polymer physics and in the study of superconductors – the filaments there are tubes of magnetic flux that poke through the sample. So actually there’s a huge range of problems involving the statistical physics of long slender filaments on many scales, polymeric, molecular or macroscopic.

**Let’s talk about your more general research. Your Wellcome Trust Investigator Award is focused on cilia, the microscopic, hairlike structures on the surface of certain cells.**

Actually it was because of my work on elastic filaments in the biological context, mainly in cell biology, that the people at Unilever thought I could contribute to the ponytail problem!

With cilia, the question is fundamentally about how they synchronise with each other in the way that they beat. This is something you see at the single cell level when you look at simple eukaryotic organisms like green algae, and you can also see it at the multicellular level when you look at cilia of all types.

And of course you see it in us because all throughout our body these hair-like appendages are beating and synchronising with each other in various fascinating ways, whether it’s perfect synchronisation or something a bit more like a Mexican wave in a sports stadium – some people raise their hands and then their neighbours raise their hands and then their neighbours raise their hands to produce a travelling wave of motion around the stadium.

**Presumably, the fluid dynamics of the environment plays a part in how they synch up?**

This is the hypothesis that has been around for several decades dating back to work of Geoffrey Taylor, a famous Cambridge fluid dynamicist back in the 1950s. He and others thought that the motion of one cilium or flagellum pushing around the fluid near it would exert a force on a neighbouring cilium or flagellum, causing it to speed up or slow down in such as way as to lead to synchronisation. But the jury’s still out – it’s only in the last four years or so that we’ve been able to test this idea using techniques such as precise high-speed imaging. Our goal is to turn the study of these systems into a real science where you can get really high quality data on wild-type organisms or mutants and test the theory in a real and quantitative way.

**You mentioned high-speed imaging. Is that the main thing you use? These ‘hairs’ are so tiny. It must be hard to isolate and see how things affect one or two neighbouring flagella, let alone control conditions to test a hypothesis.**

It is. We use the latest in high-speed imaging technology and micromanipulation and image processing methods so that we can get the most reliable data out of these videos of moving flagella. The flagella are only 10-12 microns long, a fraction of a micron across. You have to tease a good image out of a noisy raw image, then you have to apply image processing algorithms and image tracking algorithms to extract the shape of these moving objects and somehow quantify what each of two flagella are doing relative to the other. The goal is a time series of some quantity like the position of the flagellum in its orbit that can then compare that against theory.

Even then, we discovered that the beating flagella have a certain amount of intrinsic noise that comes from internal biochemistry. So you have a noisy signal with a lot of interesting dynamics on top of it and wonder: What on earth do you do to learn anything from this signal? That’s what we’ve laid the groundwork for already. The funding from the Wellcome Trust is allowing us to now really get deep physical and biological insight into what’s causing the synchronisation.

**You also study cytoplasmic streaming and growth by precipitation, like stalactites. Are these related to your main research or is it like the ponytail work?**

Some are totally unrelated! The growth by precipitation were examples of problems that came to me from a friend or someone knocking on my door saying, “I’ve found something really neat and I know you’ve got a background in a variety of different physical systems – can you help explain?” I had a great interest in pattern formation in nature before I got into biology in a big way, so I can’t resist having these side projects around natural pattern formation.

The precipitation stuff is different because the scale of the things we’re studying are larger. We work on icicles and stalactites and that’s all very macroscopic. Things are rather slow so the issue is not high speed but rather time-lapse. When an icicle is growing there’s a region of warm or cold air that is rising or falling around it and we try to visualise analyse that thin region of air using techniques from aerodynamics.

Cytoplasmic streaming is driven by molecular motors moving on filaments so it has an obvious connection with my work on flagella properties. There are plenty of things that are very different but in terms of how we study them in the lab, it’s similar.

The thing that unifies most of my work is geometry. I have a fixation on geometry and the description and motion of objects from a geometric point of view.

I studied chemistry and physics but I come from a family of architects, so maybe that’s why I have a fascination with geometry. It’s by chance that I got into the biological aspects. But I grew up in a very modern house living in a different world to my peers who lived in more conventional houses so I always thought that geometry was fascinating.

**Finally, what was it like to win an IgNobel?**

It was thrilling. When we first started this project I said, “this is the kind of thing that could win an IgNobel!”

It certainly made me very popular around the university! And since it was a pretty serious piece of scientific work in its own right, it’s had only positive connotations. When the IgNobels were created, there was a slight parody aspect to it and some people were a little uncomfortable with getting them, but there is now a real cachet to having one. It’s been overall completely positive and kind of exciting.

*Professor Raymond Goldstein is a Wellcome Trust Senior Investigator. *

Reblogged this on The Curious Optimist and commented:

Never knew statistical physics existed, but it’s quite interesting! :)