Wrapping a Perfect Sphere

The famous Austrian candy known as Mozartkugel (Mozart sphere) consists of a marzipan core that is smothered in nougat or praline cream, then coated with dark chocolate. Many companies make Mozartkugeln, but one maker, Mirabell, claims that its product is unique. It's the only Mozartkugel that's a perfect sphere.


Mozartkugeln made by Fürst. Photo by Clemens Pfeiffer, Vienna, Austria.

Each sphere comes wrapped in a square of aluminum foil. So, to minimize waste, it's natural to ask about the smallest piece of foil that would cover a sphere. It doesn't have to be a square, but it would be helpful if the unfolded shape would tile the plane. In other words, the pieces could then be cut from a large sheet of foil without any waste.

That's the tantalizing question that Erik Demaine, Martin Demaine, John Iacono, and Stefan Langerman recently bit into. They presented their findings earlier this year at the 23rd European Workshop on Computational Geometry, held in Graz, Austria.

Mathematically, the problem involves transforming a flat sheet into a positive-curvature sphere. Normally, folding (as in mathematical origami) preserves distances and curvature. Instead, the wrapping task requires infinitely many, infinitesimally small "folds" (but no stretching) to accomplish the required transformation. So, Demaine and his collaborators define a wrapping as "a continuous contractive mapping of a piece of paper into Euclidean 3-space." Contractive means that every distance either decreases or stays the same, as measured by shortest paths on the piece of paper before and after mapiing via the folding.

The analysis shows that a square with diagonal 2π and area 2π² covers a unit sphere. No smaller square can serve as a wrapping.

It's interesting to note, Demaine and his colleagues say, that a rectangle of dimensions 2π by π has the same area. It, too, can wrap around a sphere. Indeed, Mirabell wraps its Echte Salzburger Mozartkugel using such a rectangle, expanded a bit to ensure overlap.

An equilateral triangle will also do the job. In this case, the requisite triangle's area is about 0.1 percent less than the 2π² area of the smallest wrapping square.

Demaine and his colleagues also come up with a three-petal configuration (below) that tiles the plane. In this case, each wrapping unit takes up an area of only about 1.6π².


"This paper initiates a new research direction in the area of computational confectionery," the researchers conclude. "We leave as open problems the study of wrapping other geometric confectioneries, or further improving our wrappings of the Mozartkugel."

References:

Demaine, E.D., M.L. Demaine, J. Iacono, and S. Langerman. 2007. Wrapping the Mozartkugel. In Abstracts of the 20th European Workshop on Computational Geometry.

Karafin, A. 2007. "Puzzles Will Save the World." Boston Globe (June 24).