Wrapping a sphere
What is the optimal shape that can wrap a unit sphere? This question, a part of computational geometry, is still open. The paper of Demaine et al, below, establishes the new field of computational confectionery: This is because they imagine the shape to be a piece of foil, and the sphere to be a Mozartkugel.
Demaine et al set out to:
“… find shapes that have small area even when expanded to shapes that tile the plane. (They) characterize the smallest square that wraps the unit sphere, show that a 0.1% smaller equilateral triangle suffices, and find a 20% smaller shape that still tiles the plane.”
In their paper Demaine et al establish that the following petal shape -shown as part of a tiling of the plane – gives an improvement on all known wrappings of a sphere, but is still not optimal:
Their conclusion is:
“This paper initiates a new research direction in the area of computational confectionery. We leave as open problems the study of wrapping other geometric confectioneries, or further improving our wrappings of the Mozartkugel. In particular, what is the optimal convex shape that can wrap a unit sphere? What is the optimal shape that also tiles the plane? What about smooth surfaces other than the sphere?”
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. (wrapping_the_mozartkugel)
Peterson, I (2007) Wrapping a Perfect Sphere. The Mathematical Tourist: MAA.