Image depicts a representation of the Gordian Knot

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The Gordian Knot

One day, according to ancient Greek legend, a poor peasant called Gordius arrived with his wife in a public square of Phrygia in an ox cart. As chance would have it, so the legend continues, an oracle had previously informed the populace that their future king would come into town riding in a wagon. Seeing Gordius, therefore, the people made him king. In gratitude, Gordius dedicated his ox cart to Zeus, tying it up with a highly intricate knot - - the Gordian knot. Another oracle -- or maybe the same one, the legend is not specific, but oracles are plentiful in Greek mythology -- foretold that the person who untied the knot would rule all of Asia.

The problem of untying the Gordian knot resisted all attempted solutions until the year 333 B.C., when Alexander the Great -- not known for his lack of ambition when it came to ruling Asia -- cut through it with a sword. "Cheat!" you might cry. And although you might have been unwise to have pointed it out in Alexander's presence, his method did seem to go against the spirit of the problem. Surely, the challenge was to solve the puzzle solely by manipulating the knot, not by cutting it.

But wait a minute, Alexander was no dummy. As a former student of Aristotle, he would have been no stranger to logical puzzles. After all, the ancient Greek problem of squaring the circle is easy to solve if you do not restrict yourself to the stipulated tools of ruler and compass. Today we know that the circle-squaring problem as posed by the Greeks is indeed unsolvable. Using ruler and compass you cannot construct a square with the same area as a given circle. Perhaps Alexander was able to see that the Gordian knot could not be untied simply by manipulating the rope.

Commentary By: Mathematician Keith Devlin (devlin@csli.stanford.edu)