The Alexandrian–nay, Gaussian–Solution

Carl Friedrich Gauss

A year ago, I wrote about “the Alexandrian solution” to the Gordian Knot. I saw this as a metaphor for all instances in which genius lies in espying the simplicity hiding in a complex situation.

It just occurred to me that Carl Friedrich Gauss was, at the age of 10, just such an Alexander the Great. (Alexander was young, too, of course. In espying simplicity, it seems to help to be young — ie, intellectually daring, unspoiled by the complexity of life, et cetera.)

In about 1787, the young Carl Friedrich sat in class when the teacher told the kids to find the sum of the numbers 1 through 100. In other words:

1 + 2 + 3 … + 100 = ?

Think of this as the Gordian Knot. The teacher assumed that the kids would be busy for a long time, practicing their addition skills. Gauss reacted just as Alexander would have (I take poetic license):

This is too f***ing boring. There must be a simpler way.

Did Gauss get nervous as the other kids pulled ahead adding numbers, while he was still at 1, searching for simplicity? I don’t know. But he found it:

He realized that the numbers came in pairs:

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

(and so on until:)

50 + 51 = 101

So the sum of the numbers is simply (simply!)

50 x 101, or 5,050

You might, if you’re a regular on The Hannibal Blog, be guessing that I’m much less interested in sums of numbers than in, shall we say, Gordian Knots and Alexandrian Solutions in general — meaning in other, preferably surprising, walks of life.

If you can think of any instances in which daring simplicity blasted through mind-numbing complexity, drop me a line.

The Alexandrian Solution

A lot of people have a very famous story … wrong.

The story is that of the Gordian Knot and precisely how Alexander the Great loosened it. Most people imagine Alexander slashing the knot with his sword, as pictured above. But he did not.

In the nuance of how he really untied the knot lies hidden a worldview: the supremacy of simplicity and elegance over brute force and complexity. The true “Alexandrian Solution” was, for example, what Albert Einstein was looking for in his search for a Grand Unified Theory — a formula that was simple enough (!) to explain all of physics.

I’ll give you the background and the nuance of the story in a moment, but first another fist bump to Thomas for reminding us to make the association.

We are, remember, talking about complexity. The Gordian Knot is the archetypal metaphor for mind-numbing, reason-defying complexity; Alexander’s triumph over the knot is the archetypal metaphor for triumphing over complexity. Now read on…

I) Background

a) Phrygia

The Gordian Knot was, as the name implies, a knot in a city called Gordium. It was in Phrygia, an ancient kingdom in Anatolia (today’s Turkey).

The Phrygians lived near (and may have been related to) those other Anatolians of antiquity: the Trojans and the Hittites. They were Indo-European but not quite “Greek”. Their mythical kings were named either Gorgias or Midas (and one of the later Midases is the one who had “the touch” that turned everything into gold). Later, they became part of Lydia, the kingdom of Croesus. And then part of the Persian Empire. And then Alexander showed up.

b) The knot

Legend had it that the very first king, named Gorgias, was a farmer who was minding his own business and riding his ox cart. The Phrygians had no leader at that time and consulted an oracle. The oracle told them that a man riding an ox cart would become their king. Moments later, Gorgias parked his cart in the town square. In the right place at the right time. 😉

So fortuitous was this event and Gorgias’ reign that his son, named Midas, dedicated the ox cart. He did so by tying the cart — presumably by the yoke sticking out from it — to a post.

And he made the knot special. How, we do not know. But Plutarch in his Life of Alexander tells us that it was tied

with cords made of the rind of the cornel-tree … the ends of which were secretly twisted round and folded up within it.

It was a very complicated knot, in other words, and seemed to have no ends by which to untie it.

Lots of people did try to untie it, because the oracle made a second prophesy. As Plutarch said,

Whosoever should untie [the knot], for him was reserved the empire of the world.

II) Alexander, 333 BCE

Alexander, aged 23 and rather ahead of me at that age, arrived in (Persian) Phrygia in 333 BCE. The knot was still there, un-untied.

Alexander had already subdued or co-opted the Greeks, and had already crossed the Hellespont. But he had not yet become divine or conquered Egypt and Persia. All that was to come in the ten remaining years of his short life. And it began with the knot, since he knew the oracle’s prophesy.

Here he his, his sword drawn, approaching the knot:

Did he slash?

No, says Plutarch (ibid,. Vol. II, p. 152, Dryden translation):

Most authors tell the story that Alexander finding himself unable to untie the knot, … cut it asunder with his sword. But … it was easy for him to undo it, by only pulling the pin out of the pole, to which the yoke was tied, and afterwards drawing off the yoke itself from below.

III) Interpretation

I leave it to the engineering wizards among you to re-create the knot as it might have been. But what we seem to have here is a complex pattern that was nonetheless held together by only one thing: the beam.

It was, Einstein might say, like quantum physics and gravity: intimidatingly complex and yet almost certainly reducible to one simple reality.

Alexander, being Great, understood this. He saw through the complexity to the simple elegance of its solution, and pulled the peg.

This is how I understand “the Alexandrian Solution.” I intend to look for it in all of my pursuits. 😉

Bookmark and Share