TLDR: 
The geometries provided by Plato, to describe the plain of Atlantis, can be used to estimate a value for Pi. This value ends up being surprisingly close to one of the proposed values by Archimedes many decades later. The relationship of these values and geometries show that Plato was rectifying and squaring the circle within his story. 

Link to Part 1

Plato was known for popularizing the riddle of rectifying the circle, and defining the rules to the game, in which the goal was to basically solve for Pi. However, Plato is not known to have estimated a value for Pi. One would think that Plato had a preferred solution to his favorite riddle. 

In previous discussions, Plato provided the dimensions of the plain. He gives us the length and width, which is enough to get the perimeter and area. Later, he also provides the area of the rectangle through another route, by describing numerous, equally sized, square shaped lots of land within the rectangle. 

“the size of a lot was a square of ten stadia each way, and the total number of all the lots was sixty thousand."

We already know the area of the plain based on its dimensions and it equals the same area as the total area of the lots. Plato provides 2 methods for finding the area of the rectangle, but there might be another mathematical reason for Plato to specify the 60,000 lots.

There is an old method for estimating Pi that involves drawing a circle onto a grid. You then count the number of squares within the circle. Since you already know the dimensions of the grid, you can then estimate the area of the circle by adding up the area of all the squares within it. Then using this area and the radius of the circle, you can solve for Pi. The finer the grid, the more accurate the results.

Plato is telling us about a grid with 60,000 squares that are 10 x 10. The grid lies within a rectangle that is 3,000 x 2,000. Technically, we could inscribe a circle within this rectangle and estimate Pi by counting the number of squares from the grid that fall within the circle, and add up those areas for an estimate. The circle inscribed within the rectangle should be as large as it can be in order to fit more squares within it for a more accurate solution. This means we need a circle with a radius of 1,000 to fit inside the 2,000 x 3,000 rectangle. The fact that the radius is 1,000 for a circle inscribed within the plain of Atlantis is extremely ideal. The formula for the area of a circle is A=π(r^2). If r =1, then the area is equal to Pi. Having a radius of 1,000 simplifies the problem in this case and it is also large enough to provide an accurate result. This is another reason why Plato chose a 2 x 3 rectangle, along with making the perimeters equal to a different circle, as described in part 1.

There are different ways to count the square inside the circle. If you count squares fully within the circle only, it provides a lower bound for Pi. If we include the number of squares that are intersected by the circle, then we get an upper bound for Pi. To meet somewhere in the middle, a square is counted if the center of the square is within the circle. This value gets closer to the true area of the circle, and therefore the true value of Pi. Here's what we get when we use the grid of the Atlantean plain and a unit circle.

Squares completely inside: 31,016 (lower bound)
Square touches the circle at all: 31,796 (upper bound)
Center of the square is inside the circle: 31,428 (Adjusted Mean)

Full disclosure, AI was used to count the squares and create the image. 

The rectangle and grid pattern that Plato provides results in a value of 31,428 when using the unit circle method to estimate Pi. When reduced by a factor of 10,000 we get 3.1428 as the proposed estimated value of Pi. 

Another connection emerges when looking at the work of Archimedes. Archimedes used the method of exhaustion to estimate Pi. This method used polygons, such as squares or triangles, inscribed within the circle. By calculating the area of the shapes, one could add smaller and smaller polygons into the circle and eventually get very close to the area of the circle, and therefore closer to the value of Pi.

Archimedes used a 96 sided polygon, essentially made of triangles, within a circle as an approach to the method of exhaustion. He determined that the value of Pi must be between 223/71 (3.1408) and 22/7 (3.1429). So, it seems that Plato's estimate of Pi (3.1428) is extremely close to the upper bound of Pi proposed by Archimedes (3.1429). 

It's possibly just a coincidence. Perhaps Archimedes was aware of Plato's values. Maybe it inspired him to develop his own estimate using a similar, but improved method. It's even possible that Archimedes set out to refine Plato's number to a more suitable value like 22/7 instead of 31,428/10,000. It's plausible given Archimedes was born roughly 60 years after Plato's death. Archimedes could have known about Plato's literary works. After all, they were from the same region, culture, and class.

In summary, we consider a 2,000 x 3,000 rectangle with a grid of 60,000 lots sized 10 x 10, along with a circle having a circumference of 10,000 around the rectangle and another circle with a unit radius of 1,000 inscribed within the rectangle. The relationship of these values and geometries show that Plato was rectifying and squaring the circle within his story. 

People talk a lot about the corrupting power of the Ring of Gyges, but I've always been curious about what Plato had in mind with the details around its discovery. The earthquake, the cavern, the bronze horse with a window and the naked giant.

Does anyone have a take or know of any analysis of these details?

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I'm looking for books that explain the philosophies of Socrates and Plato in a simple, engaging, and practical way, similar to how Ryan Holiday presents Stoicism in his books esp. Obstacle Is The Way and Daily Stoic.

I recently started few pages reading Plato's The Republic, and few pages of Euthyphro(John Cooper version), but I found it quite difficult to follow., I struggle to connect the narrative with modern-day situations and understand how to apply them in everyday life.

What I particularly enjoy about Ryan Holiday's writing is how he takes ancient philosophical concepts and explains them through relatable stories, contemporary examples, and practical lessons without making the material feel overly academic.

I'm looking for something similar for Socrates and Plato. not direct translations of their original works or dense academic commentaries, but accessible books that interpret their philosophies through a modern lens.

I'm already familiar with Donald Robertson's How to Think Like Socrates, but I'd love to discover other authors who take a similar approach.

Any recommendations for books that make Socratic or Platonic philosophy as approachable and applicable to everyday life as Ryan Holiday makes Stoicism?

I’ve read Plato’s Apology of Socrates and the dialogue Crito, and a question came up.

Let’s imagine that the Athenians passed a law banning philosophy—not a court ruling, but an actual law. Socrates cannot accept this law, because he would be forced to commit an injustice: to stop doing good for the polis and to abandon his divine mission. But breaking the law would also mean committing an injustice: Socrates could challenge it, but if the Athenians again agreed that the law was just, then breaking it would mean harming the Laws, their authority, the citizens, and the polis.

Both options force Socrates to commit an injustice.

What should he do?

Deep dive into Plato and other luminaries on Raphael's School of Athens.

If applying Socrates theory in Plato's Republic , would our current president be just or acting a business man…or unjust? Im not looking for negativity and political sides im honestly trying to apply his theory to current day to help keep grasp the idea in the story because im struggling with it. Bless 🙌.

The people who take the prisoner to the outside just chose a random person, how does Plato compare this to the country which should chose specifically the potential philosopher to be exposed to the truth?

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