Packing Squares into a Disk with Optimal Worst-Case Density

We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is ? = 8/(5?)? 0.509. This implies that any set of (not necessarily equal) squares of total area A ? 8/5 can always be packed into a disk with radius 1; in contrast, for any ? > 0 there are sets of squares of total area 8/5+? that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square (1/2), circles in a square (?/(3+2?2) ? 0.539) and circles in a circle (1/2) have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic

How many contacts can exist between oriented squares of various sizes?

A homothetic packing of squares is any set of various-size squares with the same orientation where no two squares have overlapping interiors. If all $n$ squares have the same size then we can have up to roughly $4n$ contacts by arranging the squares in a grid formation. The maximum possible number of contacts for a set of $n$ squares will drop drastically, however, if the size of each square is chosen more-or-less randomly. In the following paper we describe a necessary and sufficient condition for determining if a set of $n$ squares with fixed sizes can be arranged into a homothetic square packing with more than $2n-2$ contacts. Using this, we then prove that any (possibly not homothetic) packing of $n$ squares will have at most $2n-2$ face-to-face contacts if the various widths of the squares do not satisfy a finite set of linear equations.Comment: 24 pages, 7 figure

Some tradeoffs in ingot shaping and price of solar photovoltaic modules

Growth of round ingots is cost-effective for sheets but leaves unused space when round cells are packed into a module. This reduces the packing efficiency, which approaches 95% for square cells, to about 78% and reduces the conversion efficiency of the module by the same ratio. Shaping these ingots into squares with regrowth of cut silicon improves the packing factor, but increases growth cost. The cost impact on solar cell modules was determined by considering shaping ingots in stages from full round to complete square. The sequence of module production with relevant price allocation guidelines is outlined. The severe penalties in add-on price due to increasing slice thickness and kerf are presented. Trade-offs between advantages of recycling silicon and shaping costs are developed for different slicing scenarios. It is shown that shaping results in cost saving of up to 21% for a 15 cm dia. ingot