U.S. mathematician Robert D. MacPherson, of The Institute of Advanced Study (Princeton, New Jersey), and U.S. physicist David Srolovitz, of Yeshiva University (New York City, New York), have discovered a mathematical formula that predicts how every single beer bubble evolves over time in a glass of beer.

Their paper, entitled “Mathematical physics: Added dimensions to grain growth” appears in the April 26, 2007 online version of the journal Nature.

The two scientists summarize their finding by saying: “Cellular structures or tessellations are ubiquitous in nature: examples include foams and crystalline grains in metals and ceramics. In many situations, the cell/grain/bubble walls move under the influence of surface tension (capillarity), with a velocity proportional to their mean curvature. As a result, the cells evolve and the structure coarsens. Over 50 years ago, the Hungarian-born mathematician John von Neumann derived an exact formula for the growth rate of a cell in a two-dimensional cellular structure. Now the much-sought extension of this result into three (or more) dimensions has been found. The formula should lead to predictive models for various industrial and commercial processes, from the heat treatment of metals to controlling the head on a glass of beer.” [Nature]

According to the authors, the math formula takes into account the change in volume of the bubble structure due directly to its surface tension. The formula involves:

([The sum of the lengths of each structure’s edge] minus [six times the mean width of the structure)]) times (a constant, that is dependent of the type of material).

What is more amazing than the perfect head on a glass of beer is that the mathematical formula can predict the expansion and contraction of individual bubbles in foam and crystalline grains in metals, semiconductors, and other such materials.

Such mathematical knowledge will help, for example, in the heat treatment of materials; specifically, as to how the tiny structures (grains) react during the process. For instance, just how the grains evolve over time can determine how strong the material becomes. Already, research on batteries that do not corrode are being performed based on MacPherson and Srolovitz’s studies, along with metal films that could easily wrap around defects and cracks.

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