Synopsis: Thinking Inside the Box

Finding the optimal solution to filling a volume with spheres could be useful for modeling nanoparticles.
Synopsis figure
C. L. Phillips et al., Phys. Rev. Lett. (2012)

Deceptively simple questions like “How many gumballs fit in a box?” or “How many stamps does it take to cover an orange?” are at the heart of some really hard problems in applied mathematics; namely, the “packing problem” and the “covering problem.” Writing in Physical Review Letters, Carolyn Phillips at the University of Michigan, Ann Arbor, and colleagues take steps to tackle filling, a new optimization problem in between packing and covering.

A two-dimensional version of the question Phillips et al. address might be posed like this: Imagine you have a square window and you want to block out as much light as possible by taping some opaque circular tiles to the glass. You can use a mixture of tiles with any radius, and they can overlap with each other, but you only have money to buy five. As a general optimization problem, this amounts to asking “What is the best way to place N overlapping circles (or, in 3D, spheres) of any size within a bounded area (volume) so as to fill it?”

The authors show that for a given shape, they only need to consider solutions where the circles or spheres lie on the shape’s medial axis—a linelike representation of the shape’s topology—and, as an example, present a numerical strategy for optimally filling 2D polygons with N circles.

Phillips et al. became interested in the filling problem as a means to help them model interactions between nanoparticles, which can be approximated as rigid bodies of overlapping spheroids. But the work could, according to the authors, be applicable to computer animation, where graphic artists look for ways to describe complex shapes with a minimum of simple, overlapping volumes. – Jessica Thomas


More Announcements »

Subject Areas

NanophysicsSoft Matter

Previous Synopsis

Materials Science

Scaling the Heights

Read More »

Next Synopsis

Quantum Information

Polarized Light in Safe Storage

Read More »

Related Articles

Synopsis: Racing to the Bottom
Fluid Dynamics

Synopsis: Racing to the Bottom

A concentrated suspension of particles can fall through a fluid faster than a single particle. Read More »

Focus: Magnetic Field Aligns Polymer Structures
Soft Matter

Focus: Magnetic Field Aligns Polymer Structures

Experiments demonstrate that a magnetic field can cause polymers to organize into arrangements that are potentially useful for clean energy products. Read More »

Synopsis: Down to Friction
Soft Matter

Synopsis: Down to Friction

Experiments explain the significant increase in viscosity that occurs when stirring a mixture of cornstarch and water. Read More »

More Articles