Scientists untangle legendary knot

Scientists untangle legendary knot


 POZNAN, Poland, April 9 (UPI) -- A Polish physicist and a Swiss biologist
may be the first to have untangled the mysteries of a legendary knot dating
back to Alexander the Great.

 Piotr Pieranski, of the Poznan University of Technology in Poland, and
Andrzej Stasiak, of the University of Lausanne in Switzerland, claim to have
discovered the mysterious and complex structure of the so-called Gordian
knot, which bound the yoke and beam of the chariot of Gordius, king of
Phrygia.

 According to legend, whoever untied the knot would become the ruler of
Asia. For centuries, pretenders to the throne tried in vain to untie the
thick rope, which became a popular tourist attraction housed in its own
temple. In 333 BC Alexander the Great is said to have cut the knot with his
sword. In so doing, Alexander defined the Gordian knot as a knot that must
be cut to be untied.

 The Gordian knot has never been fully described, Pieranski told United
Press International, though rumors have existed for centuries about its
nature. The knot was either a loop of rope with spliced ends, or a straight
length of rope with free ends, Pieranski explained.

 Pieranski told UPI that modern knot theory, a branch of mathematics called
topology, predicts that any knot tied from a straight length of rope with
free ends can be untied, regardless how complex the knot. With this
prediction in mind, Pieranski and his colleagues knew the Gordian knot must
be an "unknot," which is another name for the simplest knot, a circular loop
of rope with spliced ends.

 History supported their hypothesis. Pieranski said reports describing the
original Gordian knot, dating to 150 AD, said the ends of the rope were not
visible and so must have been spliced together.

 Starting with a loop of rope, Pieranski set out to prove his team could
construct a knot that could only be untied by cutting.

 "We answered the apparently simple question to which the answer was not
previously known: Is it possible to tie an unknot in such a way that it is
impossible to untie it and restore it back to its circular form?" Pieranski
said, adding that the simple answer is yes.

 Mathematicians view the Gordian knot problem as purely physical.
University of Massachusetts professor Rob Kusner told UPI that rope knots
are very different than their mathematical counterparts. "Mathematically, we
can't prove you can or cannot make a Gordian knot," he said from his Amherst
office.

 Eric Rawdon, assistant professor of mathematics at the University of
Pittsburgh, agreed.

 "In normal topology, knots have only one dimension and no thickness"
Rawdon told UPI. "Ropes have three dimensions, so forming a Gordian knot
from a piece of rope may be possible."

 To tie their Gordian knot, Pieranski and Stasiak used a computer algorithm
called SONO (Shrink-On-No-Overlap). SONO constructed a complicated knot by
looping and shrinking a circular piece of rope.

 Shrinking was necessary, according to Pieranski, because a loose-fitting
knot can be untied without cutting.  The rope used to make the Gordian knot
was tied and then shrunk, Pieranski said, possibly using a brine solution.

 "We propose then that the Gordian knot was a shrunken loop of rope
entangled in such a way that it could not be converted back to its original
circular form by simple manipulations," Pieranski said. Rawdon said he has
never seen such a complete description of the Gordian knot, and that
Pieranski and Stasiak have reached sound conclusions.

 Kusner agrees, though he is quick to emphasize the difference between
theory and practice.

 "The real test is to prove, mathematically, that you can or cannot
construct a Gordian knot," Kusner said. "I'm not sure a Gordian knot made of
rope rises to that standard, although I will say that Pieranski and Stasiak
have constructed an algorithm that gets stuck when it tries to untie a rope
without cutting."

 Pieranski and Stasiak are presenting their findings later this month to
the American Mathematical Society.
(Reported by UPI Science Correspondent Mike Martin in Columbia, Mo.)

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