[TDGPerson] It can't be done.

[TDGperson]

He showed that an ancient task is impossible by showing that something can never result in zero.

[AlbatrossLover]

Is the ancient task something people have done before? Claim to have done before? Something people have been trying to do for a long time?

Did he show it was physically impossible to do the task? Is it only technically impossible to do? Impossible to get the desired result from the task?

Is the "something that can never result in zero" a mathematical equation? A material that can never be depleted? Something that will always happen when people do the task?

[gregoryuconn]

Recursion relevant (e.g. it had been shown that if something could never result in zero, it couldn't result in one, if it couldn't result in one, it couldn't result in two, etc. and by showing it couldn't result in zero he showed it was impossible)?

[TDGperson]

AlbatrossLover :

Is the ancient task something people have done before? no Claim to have done before? yes Something people have been trying to do for a long time? yes

Did he show it was physically impossible to do the task? yes Is it only technically impossible to do? yes Impossible to get the desired result from the task? yes

Is the "something that can never result in zero" a mathematical equation? this one, no to others A material that can never be depleted? Something that will always happen when people do the task?

gregoryuconn :

Recursion relevant (e.g. it had been shown that if something could never result in zero, it couldn't result in one, if it couldn't result in one, it couldn't result in two, etc. and by showing it couldn't result in zero he showed it was impossible)? no

[Doriana]

Squaring the circle relevant?

[TDGperson]

Doriana

Squaring the circle relevant? yes

[GalFisk]

Did he produce a mathematical proof? Did he show that something could approach but not reach zero? Calculus relevant? Isaac Newton?

[Earnest]

Hi!

Squaring the circle relevant? yes --> so basically the task of constructing a square with the same area as a given circle by using only a finite number of steps right?
Are formulas for area of square and circle involved? so l^2 and pi*r^2

by showing that something can never result in zero --> can't the something result into 0 because it is a length? Because is a + b where b is always positive being the square of a positive number? Is Pythagorean theorem involved?

WAG --> Is the opening the compass relevant? He/she showed that the only possible instance when area of square equals area of given circle is when they are both 0. Since the opening of the compass can never be 0, then the ray of the circle is always positive and the task impossible.

By the way...now that I think about it...so the problem is l^2 = pi*r^2, which implies l = r* sqrt of pi. But pi is irrational right? So how is it possible to construct a square with side sqrt of pi? I mean...If one wanted to be precise there are tons of numbers after the comma of pi...

[TDGperson]

GalFisk

Did he produce a mathematical proof? yes Did he show that something could approach but not reach zero? no Calculus relevant? yesIsaac Newton? no

Earnest

Squaring the circle relevant? yes --> so basically the task of constructing a square with the same area as a given circle by using only a finite number of steps right? yes
Are formulas for area of square and circle involved? so l^2 and pi*r^2 yes

by showing that something can never result in zero --> can't the something result into 0 because it is a length? noBecause is a + b where b is always positive being the square of a positive number? no Is Pythagorean theorem involved? no

WAG --> Is the opening the compass relevant? He/she showed that the only possible instance when area of square equals area of given circle is when they are both 0. Since the opening of the compass can never be 0, then the ray of the circle is always positive and the task impossible. no

By the way...now that I think about it...so the problem is l^2 = pi*r^2, which implies l = r* sqrt of pi. But pi is irrational right? So how is it possible to construct a square with side sqrt of pi? I mean...If one wanted to be precise there are tons of numbers after the comma of pi... no but OTRT

[GalFisk]

Is the thing that can never result in zero a single mathematical operation? If so, addition? Subtraction? Division? Multiplication? Square? Root? Can the operation itself never give zero? Not give zero with the relevant inputs? Is pi involved in the proof?

[TDGperson]

GalFisk

Is the thing that can never result in zero a single mathematical operation?no If so, addition? Subtraction? Division? Multiplication? Square? Root? Can the operation itself never give zero? no Not give zero with the relevant inputs? yesIs pi involved in the proof? yes

[Earnest]

Is the key that it must be done in a FINITE number of steps? So the impossibility derives that it is not possible to meet the criterion of finiteness? f so, is the "never zero" part referred to the steps (e.g. if the number of steps were finite, then eventually there must be no steps left, i.e. zero steps)?

Moves to draw a square with compass and straight edge relevant? Number of moves = number of decimals after the comma of pi? Proof by contradiction?
Is the whole thing about proving that an irrational number can never be represented in a finite number of steps? I.e. because if it could than it would not be irrational...I think about something like: I start from a line of arbitrary length. Say the length is different from square root of pi first. I manage, after some manipulation to draw a line which is double (half) of it and am sure it is. Now. I am also sure that this line cannot be square root of pi otherwise I can represent it as a fraction of two numbers (sort of pi = 2* initial length). Hence I need either to add or subtract a slice to that line. If I manage to reach square of pi in a finite number of steps through this slicing/adding, then I can easily repeat the exercise starting with the line measuring sort of pi and obtain a line of double that length showing that sort of pi is not rational. Hence it is impossible to do so


Relevant how the pi was discovered? Constructing a series of squares starting from the inscribed one and ending with the circumscribed one relevant?

[TDGperson]

Is the key that it must be done in a FINITE number of steps? yesSo the impossibility derives that it is not possible to meet the criterion of finiteness?yes f so, is the "never zero" part referred to the stepsno (e.g. if the number of steps were finite, then eventually there must be no steps left, i.e. zero steps)?

Moves to draw a square with compass and straight edge relevant? noNumber of moves = number of decimals after the comma of pi? noProof by contradiction?yes
Is the whole thing about proving that an irrational number can never be represented in a finite number of steps?noI.e. because if it could than it would not be irrational...I think about something like: I start from a line of arbitrary length. Say the length is different from square root of pi first. I manage, after some manipulation to draw a line which is double (half) of it and am sure it is. Now. I am also sure that this line cannot be square root of pi otherwise I can represent it as a fraction of two numbers (sort of pi = 2* initial length). Hence I need either to add or subtract a slice to that line. If I manage to reach square of pi in a finite number of steps through this slicing/adding, then I can easily repeat the exercise starting with the line measuring sort of pi and obtain a line of double that length showing that sort of pi is not rational. Hence it is impossible to do so no, OTWT


Relevant how the pi was discovered?no Constructing a series of squares starting from the inscribed one and ending with the circumscribed one relevant?no

[Earnest]

To be sure --> do we need to assume that we have straight edge and compass? Or the number of steps can be done with arbitrary non-technological tools?

Does the proof involve other theorems? Is a system of equations involved? If so, is the system with l^2 = r*pi^2 and another equation? The equation of thee intersection between the square and thee circle (I mean, the square can be drawn everywhere in the plane but it must intersect the circle if overlapped to it)

Proof by contradiction?yes --> so do we need to start by assuming that it is possible to draw a square with side sort of pi or area = pi (which I will call assumption P) and then show that this implies that something is 0 when it can never be 0? Or we make assumption P and then proof that something can never be 0 when in reality It is?

Is the irrationality of pi involved in the proof? If so, in that it can not be represented by fractions? In that irrational numbers have relevant other properties? In that there is a theorem about them? In that they can never be represented in a finite number of steps?

Ok I am really think about it but cannot figure it out clearly. Anyway, if you agree I keep trying attacking thee problem. A way is the following. I noticed that polygons can be constructed as a combination of many squares (e.g. octagon is a combination of two squares). The idea is that one can easily move from a polygon to the a square with same area. So, if we managed to see the circle as a combination of polygons we could easily construct a square starting with the n-sided polygon. But this is impossible (i.e. the polygon should have infinitely many sides to approximate a circle)

[TDGperson]

Earnest

To be sure --> do we need to assume that we have straight edge and compass? yes Or the number of steps can be done with arbitrary non-technological tools?

Does the proof involve other theorems?yes Is a system of equations involved?yes If so, is the system with l^2 = r*pi^2 and another equation? yes The equation of thee intersection between the square and thee circle (I mean, the square can be drawn everywhere in the plane but it must intersect the circle if overlapped to it)no

Proof by contradiction?yes --> so do we need to start by assuming that it is possible to draw a square with side sort of pi or area = pi (which I will call assumption P) and then show that this implies that something is 0 when it can never be 0? yes Or we make assumption P and then proof that something can never be 0 when in reality It is?

Is the irrationality of pi involved in the proof? yes and something else If so, in that it can not be represented by fractions?this In that irrational numbers have relevant other properties? In that there is a theorem about them? In that they can never be represented in a finite number of steps?

Ok I am really think about it but cannot figure it out clearly. Anyway, if you agree I keep trying attacking thee problem. A way is the following. I noticed that polygons can be constructed as a combination of many squares (e.g. octagon is a combination of two squares). The idea is that one can easily move from a polygon to the a square with same area. So, if we managed to see the circle as a combination of polygons we could easily construct a square starting with the n-sided polygon. But this is impossible (i.e. the polygon should have infinitely many sides to approximate a circle)no, OTWT

Hint: There is a property of pi that is very relevant, other than irrationality

[Earnest]

Mmm...that it is not algebraic?!...I cannot figure out other properties of pi eithout googling so this might be my last post here xD

[TDGperson]

Mmm...that it is not algebraic?!...I cannot figure out other properties of pi eithout googling so this might be my last post here xD yes

You're very close. What is the thing that can never be zero?

[GalFisk]

Are actual fractions involved in the mathematical operation? Are fractional approximations of pi relevant?

[TDGperson]

Are actual fractions involved in the mathematical operation? yes Are fractional approximations of pi relevant? no

[Earnest]

Ohhhh maybe I see...but actually idk...so by def. algebraic numbers are non-zero numbers that make a polynomial equal to zero. Since pi is not algebraic it can never make a polynomial equal to zero. Now...I think that the polynomial in question is l^2-pi*r^2 = 0. Since the latter can never be zero it must be that l^2 is different from pi*r^2 ? Something similar? So I guess that the result is reached only asymptotically...

But actually I cannot see where fractions appear and how they are involved...

[TDGperson]

Ohhhh maybe I see...but actually idk...so by def. algebraic numbers are non-zero numbers that make a polynomial equal to zero. Since pi is not algebraic it can never make a polynomial equal to zero.That's it! Now...I think that the polynomial in question is l^2-pi*r^2 = 0. Since the latter can never be zero it must be that l^2 is different from pi*r^2 ? Something similar? So I guess that the result is reached only asymptotically...

But actually I cannot see where fractions appear and how they are involved...

***** SPOILER ******

In 1882, Ferdinand von Lindemann proved that squaring the circle is impossible by proving that pi is transcendental, which means that for any non-zero polynomial with rational coefficients, f(pi) is not zero.