Optimal packing of a finite collection of deformable objects
How should space-filling, deformable objects be packed so as to minimize the area of the interfaces between them? In 3D the search for a packing of equal-volume shapes with minimum surface area is known as the Kelvin problem. In 2D the hexagonal honeycomb is the least perimeter way to divide the plane into regions of equal area.
I will describe contributions to finite clusters of 2D objects, that is, the arrangement of N deformable “bubbles” that minimizes the total perimeter. I will describe conjectured optimal candidates (without proving anything!) under the assumption that each bubble is connected. For equal-area free clusters the candidates consist mostly of hexagons with defects confined to the perimeter. When the bubbles are confined within a fixed boundary the location of the defects becomes, in certain cases, easier to predict. For bidisperse clusters the mixing of the large and small bubbles in the optimal candidates is rather sensitive to the area ratio. And in the case of very large clusters, of more than about 5000 bubbles, extra peripheral defects are introduced to reduce the perimeter.