# New twist on ancient math problem could improve medicine, microelectronics

##### May 10, 2012

A hidden facet of a math problem that goes back to Sanskrit scrolls has just been exposed by nanotechnology researchers at the University of Michigan and the University of Connecticut.

It turns out we've been missing a version of the famous "packing problem," and its new guise could have implications for , secure , 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 tumors using fewer shots with radiation beams or speeding up the manufacturing of for .

The key to solutions in any dimension is to find a shape's "," said Carolyn Phillips, a postdoctoral fellow at Argonne National Laboratory who recently completed her Ph.D. in Glotzer's group and solved the problem as part of her dissertation.

"Every shape you want to fill has a backbone that goes through the center of the shape, like a spine," she said.

For a pentagon, the skeleton looks like a stick-drawing of a . The discs that fill the pentagon best will always have their centers 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 center, but more complicated shapes can contain multiple traps. In most optimal solutions, each trap has a disc centered 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, published online today in Physical Review Letters, 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.

Extending the approach into three dimensions, Glotzer proposes that it could decide the placement of wireless routers in a building where the signal must not be available to a potential hacker in the parking lot. Alternatively, it could help demolition workers to set off precision explosions, ensuring that the blast covers the desired region but doesn't extend beyond a building's outer walls.

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," Phillips 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.

Explore further: Researchers help Boston Marathon organizers plan for 2014 race

More information: The paper is titled "Optimal filling of shapes."
Sharon Glotzer: che.engin.umich.edu/people/glotzer.html

## Related Stories

#### Toshiba Brings HD DVD Write Drive to Desktop PCs

Jan 05, 2007

Toshiba today brought the wide ranging capabilities of HD DVD to the desktop PC with the announcement of a standard height HD DVD drive able to read and write to HD DVD and to standard DVD and CD discs. Sample shipments of ...

#### All your movies on a single DVD: study

May 20, 2009

Scientists unveiled new DVD technology on Wednesday that stores data in five dimensions, making it possible to pack more than 2,000 movies onto a single disc.

#### Exercise produces positive effects on the intervertebral discs

Jun 28, 2011

Physical exercise has a positive effect on the formation of cells in the intervertebral discs. This is shown by a study from the Sahlgrenska Academy, University of Gothenburg, presented at the annual meeting of the International ...

#### Toshiba Announces 51GB Triple-Layer HD DVD-ROM Disc

Jan 09, 2007

Toshiba Corporation today underlined the versatility and high capacity of the HD DVD format with the announcement that the company has developed a triple-layer HD DVD-ROM (read only) disc with a capacity of ...

#### Researchers shed new light on predicting spinal disc degeneration

Aug 04, 2011

About 80% of the active population suffers from low back pain at some point in their lives. In a paper published on August 4th 2011 in PLoS Computational Biology, researchers at the Institute for Bioengineering of Catalo ...

#### Entropy alone creates complex crystals from simple shapes, study shows

Dec 09, 2009

(PhysOrg.com) -- In a study that elevates the role of entropy in creating order, research led by the University of Michigan shows that certain pyramid shapes can spontaneously organize into complex quasicrystals.

## Recommended for you

#### Performance measures for CEOs vary greatly, study finds

2 hours ago

As companies file their annual proxy statements with the U.S. Securities and Exchange Commission (SEC) this spring, a new study by Rice University and Cornell University shows just how S&P 500 companies have ...

#### Ancient shark fossil reveals new insights into jaw evolution

2 hours ago

The skull of a newly discovered 325-million-year-old shark-like species suggests that early cartilaginous and bony fishes have more to tell us about the early evolution of jawed vertebrates—including humans—than ...

#### Local homicide rate increases cause more elementary students to fail school

3 hours ago

A new study finds that an increase in a municipality's homicide rate causes more elementary school students in that community to fail a grade than would do so if the rate remained stable.

#### Study: The trials of the Cherokee were reflected in their skulls

4 hours ago

Researchers from North Carolina State University and the University of Tennessee have found that environmental stressors – from the Trail of Tears to the Civil War – led to significant changes in the ...

#### Medieval slave trade routes in Eastern Europe extended from Finland and the Baltic Countries to Asia

5 hours ago

The routes of slave trade in Eastern Europe in the medieval and pre-modern period extended all the way to the Caspian Sea and Central Asia. A recent study completed at the University of Eastern Finland suggests that persons ...

#### Pioneering research unveils new method for quantifying people-related risks within the LNG industry

5 hours ago

Queensland University of Technology (QUT) has joined forces with global workforce solutions provider Air Energi Group, to develop a solution to one of the biggest challenges facing the oil and gas industry ...

## More news stories

#### Study: The trials of the Cherokee were reflected in their skulls

Researchers from North Carolina State University and the University of Tennessee have found that environmental stressors – from the Trail of Tears to the Civil War – led to significant changes in the ...

#### Ancient shark fossil reveals new insights into jaw evolution

The skull of a newly discovered 325-million-year-old shark-like species suggests that early cartilaginous and bony fishes have more to tell us about the early evolution of jawed vertebrates—including humans—than ...

#### Performance measures for CEOs vary greatly, study finds

As companies file their annual proxy statements with the U.S. Securities and Exchange Commission (SEC) this spring, a new study by Rice University and Cornell University shows just how S&P 500 companies have ...

#### Local homicide rate increases cause more elementary students to fail school

A new study finds that an increase in a municipality's homicide rate causes more elementary school students in that community to fail a grade than would do so if the rate remained stable.

#### Medieval slave trade routes in Eastern Europe extended from Finland and the Baltic Countries to Asia

The routes of slave trade in Eastern Europe in the medieval and pre-modern period extended all the way to the Caspian Sea and Central Asia. A recent study completed at the University of Eastern Finland suggests that persons ...

#### Scientists capture ultrafast snapshots of light-driven superconductivity

A new study pins down a major factor behind the appearance of superconductivity—the ability to conduct electricity with 100 percent efficiency—in a promising copper-oxide material.

#### How kids' brain structures grow as memory develops

Our ability to store memories improves during childhood, associated with structural changes in the hippocampus and its connections with prefrontal and parietal cortices. New research from UC Davis is exploring ...

#### Eavesdropping on brain cell chatter: Novel tools learn how astrocytes listen in on neurons

Everything we do—all of our movements, thoughts and feelings – are the result of neurons talking with one another, and recent studies have suggested that some of the conversations might not be all that ...

#### Gate for bacterial toxins found

Prof. Dr. Dr. Klaus Aktories and Dr. Panagiotis Papatheodorou from the Institute of Experimental and Clinical Pharmacology and Toxicology of the University of Freiburg have discovered the receptor responsible ...

#### Significant baseline levels of arsenic found in Ohio soils are due to natural processes

Geologic and soil processes are to blame for significant baseline levels of arsenic in soil throughout Ohio, according to a study published recently in the Journal of Environmental Quality.