Page 3 model: Cooking spaghetti

You there! Yes—you cooking the spaghetti! How do you tell when it’s cooked? No, don’t throw it against the wall… maths is here for you.

Cooking pasta increases the amount of water it contains until the pasta is fully hydrated, at which point we say it’s cooked. But this means that the pasta is no longer a rigid material—it’s flexible.

Nathaniel Goldberg and Oliver O’Reilly, in their 2020 paper, model the process of spaghetti cooking in a saucepan:

Measure the arclength, $s$, of your spaghetto from $s=0$ to $s=L$ and call the angle to the horizontal at any point $\theta(s)$:

Flexible rod theory tells you that the moment along the spaghetto, $\boldsymbol{m}$, depends on the angle, $\theta$, and the weight force, $\boldsymbol{n}$. For a bent spaghetto, you can write the curvature, $\kappa$, in terms of the intrinsic curvature, $\kappa_0$, and the moment, divided by the rigidity, $EI$: \[\frac{\partial\theta}{\partial s} = \kappa = \kappa_0 + \frac{m}{EI}.\]

In the pan!

But the intrinsic curvature changes as water is absorbed. Goldberg & O’Reilly use a model from plant stem growth for the rate of intrinsic curvature change, making it dependent on the amount of time the spaghetto has been in the water for:

\[\frac{\partial \kappa_0}{\partial t} = \alpha(t)(\kappa – \kappa_0) \quad \text{where} \quad \alpha(t) = \alpha_{\infty}\frac{1-\mathrm{e}^{-t/\tau}}{1+\mathrm{e}^{-(t-t_0)/\tau}}.\]

Want the spaghetti al dente? Better get working out the parameters $\alpha_\infty$ and $\tau$…


  1. NN Goldbery & OM O’Reilly Mechanics-based model for the cooking-induced deformation of spaghetti, Physical Review E, 101, 03001, (2020).

Page 3 model: Bees

If you model rabbits under ideal circumstances, you may find that the number of pairs of rabbits each month follows the Fibonacci sequence.

In this case, ‘ideal circumstances’ is a euphemism for nonsense, as your assumptions would include blatant untruths such as “rabbits mate once a month every month except their first month alive”, “a pair of rabbits gives birth to exactly one pair of rabbits per month”, and “the hutch is infinitely big (and hence Starsky is very squashed)”.

Fibonacci numbers, however, are not completely absent from nature. They accurately describe a vastly superior animal: the honeybee.

Male bees (drones) come from unfertilised eggs, and so they only have one parent — the queen.
Female bees (workers or queens) come from fertilised eggs and so have two parents — the current queen and a drone.

If you follow a drone’s family tree backwards, you will see that a drone has:

Family tree of honeybees

The number of ancestors of a male bee follows the Fibonacci sequence.

Who would’ve expected that?!


Page 3 model: Game of Thrones

Be warned: this article is dark and full of spoilers.

One of the best parts of getting into a series is getting to know and love the main characters. However, in Game of Thrones (or A Song of Ice and Fire, for you purists), this can be a heart-breaking activity. Who will survive to the end and who will bite the dust? No one knows, but perhaps maths can lend a hand.

Image: Andrew Beveridge

Andrew Beveridge and Jie Shan used network theory to investigate who the main characters of Game of Thrones are. The diagram above shows all the interactions between characters during the seventh series: the larger characters are more central, as determined by the PageRank algorithm.

However, it only takes one swing of an axe to drastically change the network…


Page 3 model: Ponytails

The ponytail hairstyle is a synonym of comfort and simplicity, and what was once considered a traditional schoolgirl style, nowadays it has become popular again thanks to clever styling. But trying to work out what shape someone’s ponytail will be has puzzled scientists and artists since Leonardo da Vinci.

In 2012, scientists from the University of Cambridge and University of Warwick developed the ponytail shape equation (PSE) to unravel some of the mysteries of the ponytail. Their model takes into account the gravity ($g$), the elasticity of the hair, the presence of random curliness of hair, and an outward swelling pressure that arises from collisions between the component hairs (which explains how a bundle of hair is swelled).

This equation can be used to find $R$, the radius of the ponytail, in terms of $s$, the arc length along the ponytail. The length at which gravity bends the hair is $l$, $L$ is the length of the ponytail, $P$ is the pressure due to the hairband, $A$ is the bending modulus, and $\rho$ is the hair’s density.

The Rapunzel number, $\text{Ra}$, of a ponytail is the ratio $L/l$. This dimensionless number determines the effect of gravity on hair. When $\text{Ra}<1$, the hair doesn’t bend much, leading to a thin, straight ponytail. When $\text{Ra}>1$, the hair bends strongly under gravity leading to a wide, bushy ponytail.

The relevance of this equation is that it could help in understanding the structure of materials made up of fiber and depicting hair realistically in animation and video games. But most importantly, if you want to look good at a party or a maths conference, simply calculate your Rapunzel number and pop on a hairband that exerts the correct pressure.


  1. RE Goldstein, PB Warren, RC Ball, Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles, Physical Review Letters, 108, 078101, (2012).
  2. RE Goldstein, (2016, September 11), Leonardo, Rapunzel and the Mathematics of Hair.

Page 3 model: Frictional unemployment

If I had a pound for every time someone assumed I studied maths because I wanted to be an economist without writing essays, I’d have enough to make it worth following the stock market. However, once the indignation fades, I can see the attraction—there are a lot of interesting uses of mathematics in economics. One of the most basic, yet most important, is modelling unemployment.

Unemployment might be caused by too few jobs in an area. Or, it may also be due to a lack of information being provided to employers or potential workers: there may be perfectly good jobs available that qualified workers simply don’t know about. This sort of unemployment is called frictional unemployment.

We split the labour force $L$ into two separate populations: employed ($N$) and unemployed ($U$). We then define $s$ and $f$ to be the rates at which people gain and lose employment:

The rate of change in unemployment is:
\frac{\text{d} U}{\text{d}t}&=\text{number becoming unemployed} -\text{number entering work}\\
If we assume that the total size of the labour force is constant, then this leads us to:
where $u$ is the proportion of the labour force that is unemployed. A lovely first order ODE, which can be solved using the integrating factor method (an exercise left for the reader). Simple enough that even an economist would understand!


Page 3 model: Crowd control

Being part of a crowd is something that we all have to experience from time to time. Whether it’s in a busy shop or commuting to work, the feeling of being swept along by those around us is all too familiar. The ubiquity of the situation, and the huge amount of data available from CCTV footage, makes crowd dynamics a favourite subject for mathematical modelling.

One popular method is known as the social force model, which applies Newton’s second law to each member of the crowd. Each individual accelerates to maintain their ‘desired velocity’, and this is balanced against forces from physical obstacles as well as the social force that maintains polite distance between people—a mathematical interpretation of personal space!

Lanes naturally form when people walk in opposite directions. Image: Dirk Helbing and Peter Molnar

Huge simulations of up to a million pedestrians have been run, which show the model’s remarkable powers. If groups of people want to travel in opposite directions along a bridge, for example, lanes of alternating direction naturally form to minimise “bumping”.

When two crowds meet at a gap, the walking direction oscillates. Image: Dirk Helbing and Peter Molnar

Some of the results are more unexpected. For example, if people try and move too fast then it can actually slow them down via an increase in ‘friction’ that results from pushing. Further, it can be shown that two narrow doors are a more effective way of leaving a room than one big door, so putting a bollard in the middle of an exit actually speeds people up!

Still, not much solace when you’re stuck in a Christmas scramble at Woolworths…


Helbing D and Molnar P (1997). Self-organization phenomena in pedestrian crowds. In: Schweitzer F (ed.) From individual to collective dynamics, 569–577.


Page 3 model

Bread is a staple of many diets. From delicious garlic bread to crunchy pizza, it’s enjoyed throughout the world. But have you ever wondered what mathematics lies just beneath the crust?  Thankfully DR Jefferson, AA Lacey and PA Sadd at Heriot-Watt University have! No? Well, we’re going to tell you anyway.


Bread dough is initially a bubbly liquid, with bubbles connected to other bubbles in a ‘matrix’.  These bubbles will collapse, provided that both the temperature and temperature gradient are high enough. To start with, the bubbles at the surface (which is hotter than the interior) reach a temperature at which they are likely to fracture. At this point, the temperature gradient is also high, with plenty of cooler liquid dough nearby. However, when the temperature of the interior has increased sufficiently to allow the bubbles inside to burst, the temperature gradient is much lower, the matrix has set, there is less liquid dough nearby, and so less collapse can take place.


But that’s not all! We can refine the model by considering the movement of the ‘crust boundary’ (where bubbles collapse) as the dough rises, as well as the vaporisation of moisture inside the bubbles. Both of these allow for the transfer of heat and affect the thermodynamics of the whole process.


So in the future, please try to remember all the maths that worked hard to ensure the crustiness of your bread! And, on that note, we’re off to get pizza…


Jefferson DR, Lacey AA & Sadd PA 2007 Crust density in bread baking: Mathematical modelling and numerical solutions. Applied Mathematical Modelling 31 (2) 209–225.
Jefferson DR, Lacey AA & Sadd PA 2007 Understanding crust formation during baking. Journal of Food Engineering 75 (4) 515–521.


Page 3 model: Hallucinations

Hallucination imageYou might think that maths and psychedelic hallucinations tend not to mix very well. But you would be mistaken! There are a series of visual hallucinations known as form constants that are highly geometric, and a mathematical model of them has provided us with some fascinating insight into how our visual cortex (the part of the brain that processes the information we receive from our eyes) works.

These hallucinations were first observed in patients who had taken mescaline, a psychedelic drug produced from a cactus found in South America. Form constants have subsequently been reported in a number of other altered states such as sensory deprivation, waking/falling asleep states, near death experiences and by individuals with synaesthesia. Some people even report seeing these patterns after closing their eyes and applying firm pressure to both eyelids for a few seconds!

Hallucination imageThe mathematical model we referred to was described in a paper by Bressloff et al., and is based on anatomical features of our brain. It seems that the visual cortex has certain symmetry properties, such as reflective, translational and even a novel shift-twist symmetry. Its electrical activity can be represented mathematically and—a bit of group theory, some eigenvectors and a couple of transformations later—has steady state solutions to the resulting equations that are remarkably similar to the observed hallucinogenic experiences. Groovy!

Disclaimer: Chalkdust does not advocate pressing hard on your eyelids.

[Written in collaboration with Samuel Mills. Pikachu adapted from picture by Matt Levya, CC BY 2.0; Hallucination pictures taken with kind permission from PC Bressloff, JD Cowan, M Golubitsky, PJ Thomas and MC Wiener, What geometric visual hallucinations tell us about the visual cortex, Neural Computation 14(3) (2002), 473–91.]


Page 3 model: Traffic flow

Have you ever reached the start of a traffic jam fearing the worst—road works, an accident, a fallen tree—to later discover no clear reason for the delay? Then you fell victim to one of the strangest traffic phenomena: the phantom traffic jam. This ghostly foe may seem supernatural, but can actually be predicted through the theory of shockwaves.

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Page 3 model: The Duckworth–Lewis method

If there are two things that typify an English summer, they are cricket and rainy days. Unfortunately, the two very often come together, which makes it very difficult to decide who should win a limited overs cricket match when rain stops play.

In these cases, a statistical model known as the Duckworth–Lewis method, devised by statistician Frank Duckworth and mathematician Tony Lewis, settles the issue (and provokes copious debate amongst Lord’s Long
Room members as they sip their champagne).

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