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Sweet Inspiration

Science  05 Sep 2008:
Vol. 321, Issue 5894, pp. 1282b-1283b
DOI: 10.1126/science.321.5894.1282b

MATHFEST 2008, 31 JULY-2 AUGUST, MADISON, WISCONSIN

Geometers find ideas everywhere. Take Mozartkugel, the famously spherical chocolate confections from Austria. Erik Demaine, his father, Martin, and colleagues John Iacono at the Polytechnic Institute of New York University and Stefan Langerman at Université Libre de Bruxelles have worked out a more efficient way to wrap them.

As mapmakers know from trying to go the other way, flattening a globe invariably distorts areas on its surface. Conversely, wrapping a globe with an inflexible wrapper (such as foil) crinkles the wrapper with infinitely many tiny folds. As a result, the area of any wrapper must exceed the surface area of the chocolate ball (4π square units for a ball with a radius of 1).

One popular brand of Mozartkugel comes in a square foil of side length π√ (π times the square root of 2). Another comes in a π ×. 2π rectangular wrapper. In each case, the wrapper's area is 2π2—some 57% greater than the surface area of the sphere. Demaine and crew set out to see if they could do better.

The computational chocolatiers found that they could achieve a 0.1% savings over current practice with an equilateral triangle whose area turns out to be approximately 1.9986π2. (The exact value for 1.9986 … is a messy formula involving, for no obvious reason, the square root of 57.) But in fact, all that really covers the kugel is a three-leaf petal inside the triangle (see figure). That means the tips of the triangle can be cut off, leaving a wrapper of area 1.8377π2.

CREDIT: A. MORALES/SCIENCE

The clipped triangular wrapper offers another advantage: The length of its perimeter, 5.3503π, is shorter than that of any other shape the researchers have found. (The square wrapper has a perimeter of 5.6569π; the rectangular one, 6π.) So a trefoil wrapper would not only save foil, Demaine and colleagues conclude, it would also be cheaper to cut. The potential reduction in the carbon footprint associated with Mozartkugel materials and manufacturing, they joke, “partially solves the global-warming problem and consequently the little-reported but equally important chocolate-melting problem.”

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