# 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…

#### References

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

You 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!

The 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!

[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.

# 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).

# Page 3 model: When zombies attack

Run? Kill? Ask nicely? (Source: flickr.com/joelf)

Every issue we feature another great model on page 3 of our magazine. This issue it’s this:

\begin{align*}
S’ &= \Pi – \beta S Z – \delta S,\\
Z’ &= \beta S Z + \zeta R – \alpha S Z ,\\
R’ &= \delta S + \alpha S Z – \zeta R.