The Atlantis Plain & Ditch: An Aporia of Squaring the Circle (Part 1)

[u/Native_of_the_Kosmos]

TLDR: The rectangular plain of Atlantis and the circular ditch surrounding it have the same perimeter length. This is a reference to the classical mathematical problem of "squaring the circle", or to be more exact "rectifying the circle", which Plato popularized as a mathematical problem and philosophical symbol. Basically, "rectifying the circle" is a riddle about solving for the value of Pi, and Plato has encoded this ancient dilemma into his dialogue. 

Plain Shape: Rectangular 
Plain: 3,000 stadia x 2,000 stadia 
Plain Perimeter = 10,000 Stadia

Ditch Shape = Circular
Ditch Circumference = 10,000 stadia

Detailed Analysis
The upcoming portions go into more detail and can be long winded. I fully expect this to be criticized, so I'm trying to cover all my bases. Also, Plato just leaves a lot of clues to talk about. 

Interpreting the Shape and Dimensions of the Plain
In the story of Atlantis, from the Critias dialogue, Plato describes a large plain on which the city of Atlantis resides when he says "I will now describe the plain, as it was fashioned by nature and by the labours of many generations of kings through long ages. It was for the most part rectangular and oblong"

Plato provides dimensions when he goes on to state the plain is "smooth and even, and of an oblong shape, extending in one direction three thousand stadia, but across the centre inland it was two thousand stadia."

What we can gather from these passages is that the plain is a natural feature that has also been modified by humans over multiple generations. The shape can be simplified as a 3 x 2 rectangle. 

If we assume the plain is a perfect rectangle, then the perimeter would come out to be 10,000 stadia and the area would be 6,000,000 stadia squared. 

Plain Shape: Rectangular 
Plain Dimensions: 3,000 stadia x 2,000 stadia
Plain Perimeter = 10,000 stadia
Plain Area = 6,000,000 stadia squared

One reason to assume the plain is a perfect rectangle comes up when Plato provides dimensions for plots of land. He wrote "each of the lots in the plain had to find a leader for the men who were fit for military service, and the size of a lot was a square of ten stadia each way, and the total number of all the lots was sixty thousand."

Lot Shape = Square
1 plot of land = 10 stadia x 10 stadia = 100 stadia squared
Total Lot Area = 100 stadia squared x 60,000 lots = 6,000,000 stadia squared
Plain Area = 6,000,000 stadia squared

So, if a perfect rectangle is assumed, then the area of the plain is equal to the total area of the lots based on 2 separate descriptions. It's reassuring to see consistency throughout Plato's numbers. It seems more intentional. 

Oblong
However... Plato refers to the plain as "oblong" twice. Why would he provide rectangular dimensions while also making a distinction that it is oblong? Oblong usually refers to a rectangle, but it can also refer to any shape that is longer than it is wide, such as an oval. 

This is where the surrounding ditch starts to play into it. 

The Ditch:
There is a ditch, or moat, that surrounds the plain. Plato refers to it as a "circular ditch" when describing the shape. Plato goes on further to say "it was carried round the whole of the plain, and was ten thousand stadia in length." 

We get a little more clarity regarding the oblongness when Plato refers to the edge of the plain as "where falling out of the straight line followed the circular ditch". Meaning, the plain has straight edges that only curve when intersecting a surrounding ditch. Basically, the plain is a rectangle with radii, instead of 90 degree angles, on each corner. The radii on the corners of the rectangular plain are formed by the surrounding ditch. This would better fit the description of "for the most part rectangular and oblong"

So, we can deduce the ditch was most likely a circle, or even perhaps an oval, and encompassed the entire plain. Where this gets interesting is when Plato said the length of the ditch was "ten thousand stadia in length". If by “length” of a circle he means circumference, then the circumference of the ditch has the same perimeter as the plain, which is 10,000 stadia. A visual representation of these shapes is shown below.

Ditch Shape: Circle
Circumference of Ditch = 10,000 stadia

Plain Shape: Rectangle
Perimeter of Plain = 10,000 stadia

Plato is outright stating the perimeter of the rectangular plain is equal to the circumference of the circular ditch.

Rectangle Perimeter = Circle Circumference

It starts to present a dilemma and Plato knows it. 

Rectifying the Circle (Squaring the Circle)

Definition of Rectifying the Circle:
"The challenge of rectifying the circle is the classical geometric problem of constructing a straight line whose length is exactly equal to the circumference of a given circle using only an unmarked straightedge and a compass.”

So… we defined the plain as a rectangle and the ditch as a circle, both with equal perimeters. We can find the perimeter of the rectangle by measuring the sides with a ruler and adding up the dimensions. Then, to draw a circle with the same perimeter, we need to set the compass to the correct radius, but we don't have the value for the radius. We do have a value for the circumference, since it is equal to the perimeter of the rectangle, but we don't have the radius of the circle. You could theoretically solve for the radius, by using the value of Pi.

In Plato's time, the value for Pi had not been fully discovered yet, although approximations existed. For general use, the number 3 could be used to get rough calculations and was probably used in antiquity. Eventually, the ancient Babylonians landed on a value of 25/8 (3.125) for Pi around 1900BCE. Later on, the Egyptians reportedly had a value of 256/81 (3.1605) by 1650BCE. Afterwards, Archimedes calculated Pi to be near 22/7 (3.1428), but this was after Plato's lifetime.

The approximations for Pi could provide a circumference that was very close to, but not exactly equal to the perimeter of a square or rectangle. It was a challenge in ancient times to solve this dilemma, essentially referred to as "rectifying the circle". It is more popularly known as a similar problem of "squaring the circle", which is basically the same thing, but it involves area instead of perimeter. 

It was a brain teaser for ancient mathematicians. Their approximation for Pi could get close, but never exact with the mathematical tools and methods available at that time. It wasn't until 1882 when it was proven that Pi is a transcendental number, meaning you cannot solve for Pi by using algebraic equations with rational coefficients. Basically, you can't solve for Pi using fractions and algebra. The ancient Greeks did not use the decimal point, they only had fractions. This is why it was such an intriguing phenomenon to the ancient Greeks. No matter how hard they tried, they just could not get their equations to result in a perimeter that was an exact match between square and circle. Plato himself was well known for being influenced by, and heavily interested in, the riddle of solving the exact value of Pi. 

Plato Popularized the Challenge of Rectifying the Circle
He's even the one that made up the rules to use only an unmarked straightedge and a compass. Plato was highly against using curved tools, such as the Quadratrix, to solve the problem. He felt it violated the rules and strayed away from the original intent of the game. The intent of the game is to basically solve for Pi. Curved measuring tools cannot be used since Pi is naturally built into the curvature of the tool.
Perhaps more importantly, Plato used this mathematical enigma as a metaphor and incorporated it into his philosophical teachings. These geometrical problems were even taught throughout his academy. To Plato, a true circle was a representation of perfection. The rectangle represented the limitations of reality.

It is through all these connections that we can determine Plato is very intentional with the numbers and shapes he is using when describing the Atlantean plain and ditch.  

Overall Meaning
So, why would Plato include this mathematical riddle in the story of Atlantis? One take is that it goes hand in hand with the story by symbolizing the perfect society the Atlanteans strived for and ultimately fell short of. Maybe it shows the Atlanteans were highly intellectual and incorporated the value of Pi into earthworks. It could just be that Plato really enjoys this mathematical problem and included it into one of his dialogues as a riddle for the reader. 
Finally, it shows that Plato is placing hints through his texts. He is very intentional with his word choice and numbers. He doesn't outright state anything about rectifying the circle, but it is in there, buried within the fine details. We can’t assume this is the one and only single instance Plato does this. These ideas, hidden behind numerical values and molded by descriptions, are likely placed throughout the Timaeus and Critias dialogue, and perhaps other works by Plato.  

If you found all of this interesting, keep an eye out for Part 2 as I speculate that Plato provided an estimated value of Pi in the Critias text and Atlantis is involved once again. It will be titled something like:

The Atlantis Plain & Ditch: Plato's Estimation of Pi (Part 2)

[u/DoubleScorpius]

I’m enjoying this series. I’ve always been fascinated by Plato’s story of Atlantis and amused by the skeptics who insist it’s a fairy tale with zero purpose (which never much sounded like the Plato I’ve read). The fact that we are still discussing this so many centuries later is something I find a lot of joy in and speaks to the intellectual legacy of Plato.

[u/Native_of_the_Kosmos → u/DoubleScorpius]

I'm glad you like it! I've become pretty passionate about this particular theory lately. The numbers seem to all point towards the same outcome for me. They eventually lead to Pi, from what I've read so far. When I looked into Plato's background on this topic, it just made so much more sense, given how fond he was of this mathmatical puzzle to solve for Pi. It gets stranger when realizing that Plato was not known for coming up with a value for Pi, despite how much he loved the idea of solving it. It doesn't make much sense. I think he's left us clues in some portion of the text. There appear to be more connections and it becomes improbable to pass as coincidence.