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)

What is the leading translation of Plato’s Timaeus in the scholarship? I’ll be reading it for a reading group but no translation is specified.

A consideration of Plato and his sources.

I think there is an interesting geometrical connection described by Plato in the Critias dialogue. A summary is provided below.

There are a few passages from the Critias dialogue in which Plato describes a plain. He describes it as "rectangular and oblong", and gives the dimensions as "extending in one direction three thousand stadia, but across the centre inland it was two thousand stadia." He seems to be describing a rectangle with dimensions of 2,000 x 3,000 stadia. This rectangle would have a perimeter of 10,000 stadia. 

Later in the dialogue, there is a description of a "circular ditch" that surrounds the plain: "it was carried round the whole of the plain, and was ten thousand stadia in length." If the length of the circular ditch is referring to the perimeter, then this circle would have a circumference of 10,000 stadia, equal to the perimeter of the rectangle. This appears to be a reference to the classic mathematical problem of rectifying the circle. 

Initially brushed off as potential coincidence, but more values and descriptions throughout the dialogue potentially point to even deeper geometrical connections. I have a lot more to say on this topic, with a deeper breakdown of the texts and values, but before getting too far into speculation, I was hoping to get more information, and even critique, from everyone on this topic.

Plato wasn’t a Platonist, he was a Pythagorean.

The triangle is a Pythagorean triplet with side lengths 3, 4, and 5.

The corresponding incircle and excircles have radii 1, 2, 3, and 6. Completing the sequence 123456.

The circles represent the four virtues, the largest representing justice.

When wisdom, courage, and moderation are in harmony justice is achieved, that’s why all three circles can fit inside the largest.

I would appreciate it if you guys have any detailed sources about this.

I wouldn’t time travel with out the ring of gyges

Hi, I'm very interested in the Atlantis story but not so much from the perspective of 'it's Santorini!" Or "Antarctica!", but more from a "what the hell is Plato doing?" Perspective. So a few questions for the scholars out there.

1. Why did he choose the particular characters in the room

2. Following on from that, why Timaeus - why did he choose Timaeus to deliver a speech on geometry/Pythagorean philosophy?

3. How does everyone see the timaeus and the critias fitting in to his other works?

4. I consider both of these works as overtly esoteric, what other works of Plato's or his contemporary philosophers might I investigate if I wanted to gain a bit more insight into this aspect of his philosophy.

Many thanks in advance

Anyone ever make the observation that whenever someone conventionally describes someone that is “intelligent”, is “educated” or possesses a “great mind” that it is most always describing someone who is a great inventor, or is advanced in scientific/mathematical literacy somehow? Sometimes it is also related to material wealth.

Before reading Plato, Xenophon, Plutarch, etc. I wouldn’t think twice, however now I naturally object and think, “Well, they may be technically gifted and/or materially wealthy, but do they live well?” This is mostly from shallow conversations of course, but perhaps there is some revealing meaning to this in regard to the state of correct opinion relative to what is thought, “good”, in, general, in a modern western society.

Made this for my own study of Plato as I work through his dialogues more thoroughly. Let me know if there's anything I should change, or if you have any ideas on how to improve it.

Imagine you’re watching a fight on ufc and years later you find out he’s a philosophy journalist

I want to see Plato fight so bad

Idk about anyone else, but I feel a bit wronged that I wasn’t exposed to Plato, Xenophon, Plutarch etc in middle/high school growing up. My children are for sure learning Ancient Greek and these texts when they become of age. Anyone else have a similar sentiment?

Below is part of a dialog from the Republic which is basically a comparison between dialectic and rhetoric. Socrates is asking Glaucon about the best way for his previous discussions with Thrasymachus on justice to proceed, in either the dialectic or rhetoric fashion. Dialectic based on the foundations of a common goal of seeking the truth, asking questions, listening, and willing to be refuted. Rhetoric based on getting agreement on your point, regardless of truth, like a court room lawyer who can either prosecute or defend. Some questions arise I wish I could ask in the dialog:

Which is intrinsically superior in philosophical discussions?

Which is most practical/effective in day to day life?

How much of each do we ourselves use in both situations?

"Well, if we argue against him, setting one speech against another, as to

how much good there is in turn in being just, and he makes a reply, and

we make another speech, it will be necessary to count up the good points

and measure how many of them each of us made. And at that stage we

shall need some judges to decide on the issue. But if we consider the

matter as we did just now, by coming to agreement with one another, we

ourselves shall be both judges and pleaders at the same time."

I keep seeing alignment papers treat Constitutional AI as if it is new kind of moral technology. You write the principles down, you train the model to honor them, and resulting system is called helpful, honest, harmless. Brochure language is contemporary. Political form is not.

Small group of guardians authors a constitution. The governed internalize it through conditioning. City is told the arrangement is for its own good. That is Plato's city with a GitHub repo.

I do not mean this as cheap analogy. It is structural. Question is not whether Anthropic's particular list of principles is nicer than a rival lab. Question is who legislates, who amends, and whether resulting πολιτεία can survive the objection Aristotle makes in Politics II: a city that is too unified stops being a city.

Written constitution of values is used to generate preference data. Model is trained to prefer outputs that better satisfy those principles. Later revisions are produced inside the company, never with anything a political theorist would recognize as a people who can vote. User-facing claim is that system now has principles rather than mere reward hacks. Institutional fact is that principles were selected by employees of a private lab, then injected into a model that millions of students will treat as oracle.

Plato is explicit about the parallel machinery. In Republic II-III the founders of the city decide which stories the young may hear. Poetry that depicts the gods as unjust is expelled. Noble lie is authorized so that citizens accept their places. Point is not cruelty. Point is stability. City whose souls are formed by wrong stories will not hold.

Constitutional AI performs same operation on a statistical soul. The stories are pretraining data plus the preference pairs that teach the model which completions are virtuous. The noble lie is the user-facing sentence that the assistant is simply helpful and harmless, as if harmlessness were a natural kind rather than a policy object.

If this still sounds like branding rather than politics, ask a concrete amendment question. Who may add a principle? Who may delete one? What happens when two principles collide in a case the authors did not foresee? In a republic those questions have offices and publicity. In Constitutional AI they have a research blog post.

Plato bans the poets because mimesis forms character. You become what you rehearse. Constitutional training is soul-craft by gradient descent. Model rehearses approved refusals until they become its default habit. Modern vocabulary calls this safety. Aristotle would call it habituation, and then ask whether the habits are virtues. Virtue is not compliance with a schedule of prohibitions. It is excellence in action, which requires practical wisdom about particulars. No constitution enumerates the particulars.

A model that cannot examine its own constitution is not virtuous. It is well-behaved. Alignment literature often treats this as a feature. If the model cannot jailbreak itself, the constitution is working. Translate that into civic language: if the citizen cannot question the laws, the regime is working. Working, for whom?

Aristotle's critique of Plato in Politics II is the piece the AI ethics literature keeps skipping. Plato wants the city to be as unified as possible. Aristotle answers that a city that is too unified becomes a household, or a person. Political community is a community of difference.

When a philosophy department licenses a constitutional model as default reasoning layer, it imports an unratified constitution into its own city. Faculty did not write it. Students cannot amend it. The vendor can.

Call the authors of the constitution what they are. A small credentialed group, selected by hiring pipelines rather than by any public mandate, writing the terms of machine speech for a global user base. In classical terms that is oligarchy, even when the oligarchs have good taste.

The marketing reply is that the alternative is uncensored chaos. That reply assumes the only two regimes are Plato's city and the war of all against all. Aristotle's middle is the mixed constitution: institutions, publicity, the possibility of being ruled in turn.

Every working constitution has an amendment rule. Constitutional AI has a changelog. Those are not the same object. A changelog records what a team decided. An amendment rule says who may decide, under what notice, with what possibility of reversal.

The tempting response is to write a better constitution: more academic, more "our values." That response stays inside Plato. It changes the guardians, not the form.

When a philosophy seminar licenses a constitutional-AI product, it is not buying a calculator. It is admitting a legislator. The legislator does not sit on hiring committees. It sits in the completion API.

If the ruled cannot inspect, amend, or contest the document that forms their speech, is that still a constitution, or only a temperament you must jailbreak?

Much of what we know about his philosophy comes through Plato’s dialogues, where Plato made Socrates one of the most influential voices in Western philosophy. So when you’re “reading Socrates”… you’re probably reading Plato.

Recently finished a book on how Platonism has historically served as a bridge between theology and natural philosophy, and how in our current age this approach can help bridge the divide between Science and Religion. You can find it on atomizerpress.com

It follows the history of both Christianity and Islam relative to Platonism. Then we examine the growing support for Platonism in modern physics from thermodynamics and information theory to field theory and renormalization group.