Ancient Math Problem Could Improve Medicine

Photo: Math formula

It turns out we've been missing a version of the famous "packing problem," and its new guise could have implications for cancer treatment, secure wireless networks, microelectronics and demolitions, the researchers say.

Called the "filling problem," it seeks the best way to cover the inside of an object with a particular shape, such as filling a triangle with discs of varying sizes. Unlike the traditional packing problem, the discs can overlap. It also differs from the "covering problem" because the discs can't extend beyond the triangle's boundaries.

"Besides introducing the problem, we also provided a solution in two dimensions," said Sharon Glotzer, U-M professor of chemical engineering.

That solution makes it immediately applicable to treating tumours using fewer shots with radiation beams or speeding up the manufacturing of silicon chips for microprocessors. The key to solutions in any dimension is to find a shape's "skeleton," said Doctor Carolyn Phillips.

"Every shape you want to fill has a backbone that goes through the centre of the shape, like a spine," she said. For a pentagon, the skeleton looks like a stick-drawing of a starfish. The discs that fill the pentagon best will always have their centres on one of those lines. Junctions between lines in the skeleton are special points that Glotzer's team refers to as "traps." The pentagon only has one trap, right at its centre, but more complicated shapes can contain multiple traps. In most optimal solutions, each trap has a disc cantered over it, Phillips said.

Other discs in the pattern change size and move around, depending on how many discs are allowed, but those over the traps are always the same. Phillips suspects that if a design uses enough discs, every trap will have a disc centered over it. In their paper the researchers report the rules for how to find the ideal size and spacing of the discs that fill a shape. In the future, they expect to reveal an algorithm that can take the desired shape and the number of discs, or the shape and percentage of the area to be filled, and spit out the best pattern to fill it.

Phillips expects filling solutions to be scientifically useful as well. Glotzer's team developed the new problem by trying to find a way to represent many-sided shapes for their computer models of nanoparticles. In addition to nanotechnology, biology and medicine often need models for complex shapes, such as those of proteins: “You don't want to model every single one of the thousands of atoms that make up this protein," she said. "You want a minimal model that gives the shape, allowing the proteins to interact in a lock-and-key way, as they do in nature." The filling approach may prove a perfect fit for a variety of fields.; Source: University of Michigan