Steinhaus solution (for the seriously math minded)

Mysterious Math

Math E Matics

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Christopher Columbus helped prove the world is round. Jacques Cousteau discovered the secrets of the deep sea.

Dan Mauldin and Steve Jackson never sailed a ship, nor charted ocean depths. Yet they are world-class explorers.

Their mission is to solve mathematical mysteries. They recently ended a four-year quest to find the answer to one problem that had stumped math whizzes worldwide for more than 50 years.

"Solving a math problem is like going where no other mathematician has gone before," says Jackson. "You seek a solution because you’re curious and because it might lead to great discoveries."

The problem

The two UNT math professors undertook their journey after Alekos Kechris, a mathematics professor at the California Institute of Technology, posed a question.

 Math professors Dan Mauldin (left) and Steve Jackson solved the Steinhaus lattice problem, which had stumped math whizzes for more than 50 years.

"One of my colleagues was writing about the Steinhaus lattice problem," says Kechris. "I asked Dr. Mauldin during a math conference if he knew anything about the problem."

Polish mathematician Hugo Steinhaus first posed it in the 1950s.

The problem starts with basic geometry. It involves a pattern of points, a lattice and a flat two-dimensional plane.

Steinhaus asks if a set of points on the plane can be created, where exactly one point in the set will match one intersection on a lattice — no matter how the lattice is moved.

Jackson and Mauldin created such a set. Their solution might be understood by looking at the night sky.

Imagine the Steinhaus set of points is a constellation of stars. The lattice is a grid much like a wire window screen.

Now, throw the grid against the night sky. Only one star in the constellation is captured at one intersection of the grid. Repeat the process by throwing the lattice up in another direction.

Each time you throw the lattice screen up, exactly one star in the constellation is always captured by one intersection on the grid.

The partnership

Both Mauldin and Jackson agree that finding a solution to the Steinhaus problem required much more than math skills. Not only did they use their intellects to master a math riddle, but they also employed their wits to overcome doubt.

"It takes a lot of courage to explore an area where no one else has been before," Mauldin says.

Both mathematicians thought the problem was simple enough and that they were on the right track to solving it, but there were times when they got "stuck."

"Imagine finding out you’re not on the right path," Mauldin says. "First, and most importantly, you must face the facts. You have to admit you’re going in the wrong direction. Once that is done, then you can go back and salvage what was right about what you were doing."

He says it takes psychological strength to experience these glitches time and time again and it was during these setbacks that he and Jackson really counted on each other for support.

"It was our 'foolish enthusiasm' that got us through rough spots," Mauldin says.

Once they solved the Steinhaus problem, Mauldin and Jackson sought out new problems for which they could exercise skills and techniques they had developed.

According to Jackson, a solution leads to the next problem and then to the next solution. He says all of math is interconnected.

Mauldin agrees: "It’s the legacy of math. In the tradition of Archimedes, Euclid and Newton, one mathematician passes knowledge to the next generation of mathematicians. We continue to explore possibilities and seek answers."

He says they used results from numbers theory discovered 200 years ago to solve the Steinhaus problem. Those results had no practical applications at that time, but today, they help make computer communications possible and secure.

"Today, the Steinhaus problem is a theoretical exercise, but one day, it could have far-reaching effects in solving future problems," he says.

Like Columbus and Cousteau, Mauldin and Jackson have charted new territory.

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