The enemy is at the gates, oh wait that was yesterday’s prompt. We often encounter difficult, seemingly unsolvable problems and many mathematicians like to say that Math does not have problems, it only has solutions. Others may try to trivialize problems by calling them challenges. This is often done to motivate students to keep working. But when we are at war, solutions must be found, as the enemy will use all of its cunning and guile to develop a superior strategy, so they can annihilate us. We must put on our thinking caps (or perhaps Phrygian caps) and put our best foot forward, rise to the challenge and get the solution.
There is a Greek legend about a man named Gordius (the father of the celebrated King Midas), who was a poor peasant that had just happened to drive into town of Phrygia one day with his wife in an ox cart. The king of Phrygia had recently died unexpectedly and the Phrygians temporarily found themselves without a legitimate king, just before Gordius arrived in Phrygia. The oracle at Telmissus, which was the ancient capital of Phrygia had informed the residents that their future king would enter their city driving an ox-cart. They were all delighted upon seeing Gordius, and the people declared him to be their king. Showing his gratitude, Gordius dedicated his ox cart to the Phrygian god Sabazios, who was the nomadic horseman and sky father god of the Phrygians and Thracians, being sort of an equivalent to Zeus himself. To keep this cart from being tampered with, Gordius tied it up with an extremely intricate knot that is now known as the Gordian knot. Upon inspecting the knot the oracle made another prediction, stating that the person who was able to untie this knot would rule all of Asia.
Yea, I want that as my superpower, being able to see the future. It seems like any Greek story worth telling always featured an oracle or two, but their days are long gone. I don’t think that there was one on every corner, but perhaps each town had its own oracle. Meeting with an oracle would allow the Greeks to communicate with the gods and this usually only happened in specific places, and at certain times (possibly after the oracle was paid). Through the ability of the oracle, people were united with the gods in their thoughts, so they could obtain advice and learn what was going to happen in the future.
The Gordian Knot is often used as a metaphor for any troublesome, demanding or otherwise unrelenting problem, such as disentangling an impossible knot, that might be easily solved by cheating or what we refer to today as thinking outside the box. This tied up ox-cart was viewed to be an emblem of power and constant military readiness to the Phrygians. Many fine minds had been stumped by the Gordian knot, and it still stood laying dormant at the gate to the city to discourage enemies of Phrygia for several generations. It resisted all attempted solutions and then Alexander the Macedonian conqueror arrived to seize their city, and make it part of his Empire.
Alexander realized that he could avoid any conflict with the Phrygians if he was just able to untie this knot. The impetuous Alexander was instantly seized with an ardent desire to untie the Gordian knot, so he did a thorough examination and he was not able to find any free ends on this knot. Alexander was a former student of Aristotle, and he probably learned how to solve many puzzles in his day. He first thought that this was made up of several knots, all so tightly entangled that it was impossible to see how they were fastened. Alexander reasoned that two ends of the rope must have been spliced together, thus this Gordian knot could not be untied simply by manipulating the rope. In order to unbind it, he sliced it in half with a stroke of his sword, which is called the Alexandrian solution, a method where bold or violent means is used instead of rational thinking. Alexander proclaimed, “It makes no difference how this knot is loosed”, and after slicing the knot in half, he went on to conquer Asia, fulfilling the prophecy. Cutting the Gordian knot has become a phrase used when a bold solution is used to solve a complicated problem.
Did Alexander cheat, was the challenge of solving this puzzle to be done solely by manipulating the knot, and not by cutting it? Was this Gordian knot even a knot, or was it constructed from some sacred geometry, or was it possibly a simple torus? A torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle and it looks like a donut. Maybe the rope was thoroughly wetted prior to its being tied, then after tying the knot, it was allowed to dry in the sun which made it shrink.
Perhaps the Gordian knot was constructed by first splicing together two ends of a length of rope to form a loop, and then tying the loop up or wrapping it around itself in some way, to disguise the fact that it was not really a knot at all. Making this assumption, the Gordian knot was really just a loop, being impossible to manipulate and therefore it would have been absolutely necessary to resort to a sword to untie it. Modern topologists study knots, assuming that knots are constructed out of perfectly flexible, perfectly stretchable, infinitely thin string.