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# The Incredible, Edible Foam--and the Mysterious Mathematics behind It

Mathematicians are still struggling to understand what happens atop your morning cappuccino

Image: Ryan Smith Modernist Cuisine, LLC

If you sometimes start your morning with a frothy cappuccino and finish off the evening  with a heady glass of beer, then your day opens and closes with one of the most scientifically intriguing kinds of food: the edible foam. There are deep mathematical mysteries in these interlocked bubbles, and recently they have also become one of the most fertile areas for culinary innovation.

Top-ranked chef Ferran Adrià of elBulli in Catalonia, Spain, began experimenting with culinary foams in the mid-1990s in his quest to present diners with new and unexpected culinary experiences. Adrià used unconventional foaming agents such as gelatin or lecithin rather than eggs or cream. He used whipping siphons pow­ered by pressurized nitrous oxide—much like cans of Reddi-wip but sturdier—to create eth­er­eal foams from foods as diverse as cod, foie gras, mushrooms and potatoes. That started a revolution in foams, as chefs, among them Hes­ton Blumenthal of Bray, England, New York City’s Wylie Dufresne, and Chicago’s Grant Achatz, have taken to foaming all manner of savory foods.

These dishes have an aura of mystique about them and not just for their novel texture. Although foams may look like random jumbles, the bubbles within all foams seem to self-organize to obey three universal rules first observed by Belgian physicist Joseph Plateau in 1873. These rules are simple to describe but have been remarkably hard to explain. The first rule is that whenever bubbles join, three film surfaces intersect at every edge. Not two; never four—always three. Second, each pair of intersecting films, once they have stabilized, forms an angle of exactly 120 degrees. Finally, wherever edges meet at a point, the edges always number exactly four, and the angle is always the inverse cosine of –1/3 (about 109.5 degrees).

Only a century later, in 1976, did Rutgers University mathematician Jean Taylor prove that, at least in the case of two joined bubbles, Plateau’s rules derive from the action of surface tension, which forces the bubbles to adopt the most stable configuration. Mathematicians are still attempting to nail down exactly what happens in a froth of three or more bubbles, as well as the unsolved question of what arrangement of bubble shapes in a foam will fill a container while using the least surface area (and thus the least energy). In 1887 Lord Kelvin had proposed that a honeycomb of tetra­dec­a­hedrons, each with six square and eight hexagonal faces, is the answer. But in 1994 phys­i­cists Dennis Weaire and Robert Phel­an of Trinity College in Dublin published an even better—though not neces­sar­ily optimal—solution: a foam of two kinds of cells, one made solely from 12 pentagons and the other constructed from two hexagons and 10 pentagons.

In foamy foods, bubbles that do not follow Plateau’s rules quickly pop. The same fate occurs to bubbles that are too small: surface tension raises the pressure inside them beyond the breaking point. That is one reason that liquid foams become coarser as they age—and why it is best to sip your cappuccino while it is fresh.

This article was originally published with the title The Incredible, Edible Foam.

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Myhrvold is author and Gibbs is editor of Modernist Cuisine: The Art and Science of Cooking, scheduled for publication in March 2011.

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1. 1. agenthucky 12:59 PM 11/30/10

I doubt I will find an expert here, but being a home brewer, getting a good foam (head) is quite important IMO. There are certain types of ingredients that help the retention of the foam. Some of these include high protein grains. When the proteins get left in the beer, they stablize the foam, giving it a structural advantage, so it lasts longer.

I wonder what sort of studys they did with not only optimal surface tension, but adding structural ingredients to maximize the foams lifespan.

2. 2. reflectogenesis 02:42 PM 11/30/10

Could the optimal structure of a foam depend upon the difference between advancing and receding contact angles.
So at the point of stability - the structure adopted relates to the maximum net rate of change of surface and cohesive energy densities away from the equilibrium structure.

3. 3. Matthew Roy in reply to agenthucky 02:50 PM 11/30/10

Aren't bubbles great... It is my understanding that reasonably stable foams result when water is mixed with a solute that has a polar and non-polar component to the molecule. The simplest example is a phospholipid. The polar (hydrophilic) end of the molecule attracts water, and the non-polar (hydrophobic) end of the molecule tries to get away from the water. The result is a sphere (lowest energy configuration) with the hydrophobic ends of the molecules in the centre and the hydrophilic ends on the outside. This effect is what makes cell membranes and soap bubbles.

Another major factor in making a good head is the nature of the air entrained in the pour. I'm sure you're well aware that depending on how you pour a beer determines the size and the nature of the head. The pour is the regulation of the size and number of air pockets being entrained in the entering flow.

So in answer to your question, it would depend on the protein. Proteins generally have this polar / non-polar property, but the amount of charge imbalance depends on the protein. Some form stable strands (alpha helices), sheets (beta strucutres) or twisted balls (tertiary or globular) in water. Simpler proteins probably make for better bubbles.

I'm not sure that science can help you make a better head on your beer, but a hint may be in the concentration of whatever proteins are in your beer. You will note that when blowing bubbles, the mix of soap to water determines the size, stability and longevity of the bubbles. A very tough mathematical problem to say the least.

4. 4. irenealhanati in reply to Matthew Roy 07:42 PM 11/30/10

I enjoyed so much reading your comments!I have never enjoyed mathematics, but after reading your comments I think I missed a lot! I´m an art teacher and I´m fond of studying the connection between art and science. Thanks for this class!

5. 5. Quinn the Eskimo 10:32 PM 12/3/10

Foam in a can?

Redi-whip. Cheap. Available. Not new science.

6. 6. wayt in reply to agenthucky 07:58 PM 12/14/10

Stay tuned! Our next column in the January issue talks about beer foams and how they are stabilized.

7. 7. wayt in reply to Matthew Roy 08:07 PM 12/14/10

For those who are interested in reading more about everyday foams and the amazing mathematics, physics, and chemistry involved in them, I strongly recommend the book "Universal Foam" by Sidney Perkowitz. It's a fun and easy ready. http://www.amazon.com/Universal-Foam-Exploring-Mysterious-Substance/dp/038572070X

And of course our book, coming out next spring, also has a lot more discussion, photos, illustrations, and recipes in its chapter on Foams. http://modernistcuisine.com/

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