
Steinhaus solution (for the seriously math minded)
Mysterious
Math
Math
E Matics
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Spirit Bell



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 worldclass explorers.
Their mission is to solve mathematical mysteries. They recently ended
a fouryear 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.
"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 twodimensional 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 farreaching effects in solving future problems," he
says.
Like Columbus and Cousteau, Mauldin and Jackson have charted
new territory.

