# The mathematical link: snowflakes and teeth

A formula used by mathematicians and physicists to describe the formation of snowflakes is proving equally useful in predicting the distribution of enamel across teeth – a finding with implications for fields as diverse as dentistry and palaeontology.

Understanding how enamel becomes distributed across the crown of a tooth, and how it varies between species, has been a persistent challenge. The material is quite remarkable, in that it starts out rather soft and then undergoes a phase change to become solid – indeed, one of the hardest substances in the world, able to withstand ageing and fossilisation for millions of years.

Finding the factors that drive the specifics of enamel distribution would go a long way towards explaining why, for instance, orangutans and humans, despite being closely related and being broadly similar in shape, have very different external tooth-shapes.

Researchers from University of Helsinki and Aalto University, both in Finland, hypothesised that the distribution was controlled by nutrient diffusion – specifically, the rates at which different areas of a tooth receive the nutrients required to build enamel.

To test the idea, they turned to a well-supported mathematical model called the Stefan Problem, which is used to describe phase transitions in matter.

Specifically, it aims to describe the temperature distribution in a uniform medium that is undergoing a phase change. It treats moving phase boundaries as expressions within a partial differential equation.

All of which means that it is a very good method for mapping, and predicting, the formation of crystals, including snowflakes.

To see if the same model could be applied to enamel formation, lead researcher Teemu Häkkinen and colleagues used computed tomography to gain images of real teeth, from which the existing enamel was later digitally removed.

They then fiddled with various parameters and ran a rebuild program using a Stefan model, to see what factors influenced the way enamel spread, shaped and hardened.

When they used variables that limited the diffusion of nutrients, they were able to reproduce the proper thickness and contours of a human molar.

“Whereas enamel is not obviously as intriguingly shaped as snowflakes, it is interesting that the same physical principles can account for the increase in complexity in both systems,” says Häkkinen.

The researchers then repeated the experiment, using orangutan and pig teeth. In both cases, a nutrient limiting matrix produced exact likenesses.

“There are huge amounts of different data available on enamel, and now we have the tools of physicists to make testable predictions,” says co-author Jukka Jernvall.

The research is published in the journal PLOS Computational Biology.