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Proof that pi=4

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Thevoiceofreason

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Proof that pi=4
« on: March 13, 2011, 06:50:27 AM »


This is obviously false, but does anyone have any disproof, other than proving that pi is approximately 3.14159265358979...?

My initial response was that of fractal dimensions. I don't think you can ascribe a length to this construction in the usual manner, as fractals tend to be 1.x dimensional.

Another, more grounded disproof is through calculus. The derivative as any given point on a circle, in Cartesian coordinates is non-trivial, but solvable or at least defined. However, if the infinite triangles case, the derivative doesn't exist anywhere, as it is not mathematically smooth in the limit sense.

Thoughts?
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Thork

Re: Proof that pi=4
« Reply #1 on: March 13, 2011, 08:43:50 AM »
Squaring the circle huh? Its a very old problem, you can start here
http://en.wikipedia.org/wiki/Squaring_the_circle
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Parsifal

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Re: Proof that pi=4
« Reply #2 on: March 13, 2011, 01:33:34 PM »
Quote from: Thork on March 13, 2011, 08:43:50 AM
Squaring the circle huh? Its a very old problem, you can start here
http://en.wikipedia.org/wiki/Squaring_the_circle

No, that isn't what he's doing.

Anyway, the flawed step is the "repeat to infinity" bit. Infinity is not a number, and no matter how many times you repeat this process, you will not perfectly match the circle -- you will only get very, very close to it.
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Thork

Re: Proof that pi=4
« Reply #3 on: March 13, 2011, 02:26:26 PM »
Actually he is doing what I said (area is directly proportional to dimensions of a shape), but I will make it simpler for you.

Imagine having a piece of string 3.14 blah blah metres long. And you fuse it together at the ends and put it in a circle so the diameter is 1m. Now try pulling the first corner of the square out of it, to make the square. You can't because you need a longer piece of string. What you have done to make it fit the 3.14 circle as you tend towards infinity is infinitely fold the perimeter back on itself so it follows the 3.14 metre outline, but you have enough folds to make it still be 4m long.

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PizzaPlanet

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Re: Proof that pi=4
« Reply #4 on: March 14, 2011, 03:31:09 AM »
He's not squaring a circle, and I have no clue where you got that idea from.
Also, saying "infinity is the flaw here" isn't very convincing. We'd still have to agree that pi tends to 4, which it doesn't.
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Parsifal

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Re: Proof that pi=4
« Reply #5 on: March 14, 2011, 04:12:04 AM »
Quote from: PizzaPlanet on March 14, 2011, 03:31:09 AM
Also, saying "infinity is the flaw here" isn't very convincing. We'd still have to agree that pi tends to 4, which it doesn't.

False.
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Quote from: Ævan on August 11, 2013, 01:18:20 PM
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PizzaPlanet

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Re: Proof that pi=4
« Reply #6 on: March 14, 2011, 04:12:45 AM »
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Parsifal

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Re: Proof that pi=4
« Reply #7 on: March 14, 2011, 04:14:42 AM »
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Quote from: Ævan on August 11, 2013, 01:18:20 PM
I'm going to side with the white supremacists.
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Saddam Hussein

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Re: Proof that pi=4
« Reply #8 on: March 14, 2011, 05:02:56 AM »
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Marcus Aurelius

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Re: Proof that pi=4
« Reply #9 on: March 14, 2011, 08:25:39 AM »
What happens if d=25?  Wouldn't that make pi 100 according to that math?  ???
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ﮎingulaЯiτy

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Re: Proof that pi=4
« Reply #10 on: March 14, 2011, 08:42:19 AM »
Quote from: Marcus Aurelius on March 14, 2011, 08:25:39 AM
What happens if d=25?  Wouldn't that make pi 100 according to that math?  ???

Erm, the false circumference would be 100.
Pi is circumference divided by diameter, but that second part is invisible while the diameter is 1. It maintains the ratio of 4 from the relation of the square.
« Last Edit: March 14, 2011, 08:44:38 AM by ﮎingulaЯiτy »
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Username

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Re: Proof that pi=4
« Reply #11 on: March 14, 2011, 03:36:28 PM »
Looking at this briefly, I recall the diagonal paradox.
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Trekky0623

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Re: Proof that pi=4
« Reply #12 on: March 14, 2011, 04:10:37 PM »
You're making up and down vertical motions as opposed to diagonal motions. So you're basically increasing the distance around your turns of the circle, and increasing the circumference. A better approximation is increasing polygons.
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Trekky0623

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Re: Proof that pi=4
« Reply #13 on: March 14, 2011, 04:21:21 PM »
For example, the "octagon" made out of squares above has a perimeter of 4, but a true octagon would have a perimeter of 3.0615, so obviously these "pixel shapes" are not very accurate.
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Thevoiceofreason

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Re: Proof that pi=4
« Reply #14 on: March 14, 2011, 11:27:26 PM »
Quote from: John Davis on March 14, 2011, 03:36:28 PM
Looking at this briefly, I recall the diagonal paradox.

yeah this is the root of the problem.

If they were the same construction, then we'd have a problem with the laws of mathematics.
The simplest way to show that they aren't, is the derivative argument.

And Parsifal's argument has some validity to it. You cannot use a finite process of arbitrary length to imitate infinity. Like the proof that pi is rational, based on the fact that pi to x digits is rational for any x.
Perhaps the other points (total length=4-pi) lie in a fractal dimension.
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EnglshGentleman

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Re: Proof that pi=4
« Reply #15 on: March 15, 2011, 09:27:02 PM »
From what I have been able to find out, this is suppose to be a classic difference between "Euclidean Norm" and "Taxi-cab Norm" which means Pi may not even be a constant as it applies to the problem.

I am too dumb to have a clue what that means though.  :(
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ﮎingulaЯiτy

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Re: Proof that pi=4
« Reply #16 on: March 15, 2011, 11:56:52 PM »
Quote from: EnglshGentleman on March 15, 2011, 09:27:02 PM
From what I have been able to find out, this is suppose to be a classic difference between "Euclidean Norm" and "Taxi-cab Norm" which means Pi may not even be a constant as it applies to the problem.

I am too dumb to have a clue what that means though.  :(

Assuming you went to Wikipedia, I dare say you shouldn't read about Taxi-cab Norms on the page for mathematical norms.
Did you see this page?   http://en.wikipedia.org/wiki/Taxicab_geometry
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PizzaPlanet

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Re: Proof that pi=4
« Reply #17 on: March 15, 2011, 11:58:00 PM »
Quote from: EnglshGentleman on March 15, 2011, 09:27:02 PM
From what I have been able to find out, this is suppose to be a classic difference between "Euclidean Norm" and "Taxi-cab Norm" which means Pi may not even be a constant as it applies to the problem.

I am too dumb to have a clue what that means though.  :(
Nah, John tackled it perfectly. The issue lies in the fact that we keep the perimeter constant, aka we're not approaching a circle at all, despite of how optically convincing it might be.
« Last Edit: March 16, 2011, 01:53:19 AM by PizzaPlanet »
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ﮎingulaЯiτy

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Re: Proof that pi=4
« Reply #18 on: March 16, 2011, 12:01:14 AM »
Quote from: PizzaPlanet on March 15, 2011, 11:58:00 PM
Nah, John tackled it perfectly. The issue lies in the fact that we keep the perimeter constant, aka we're not approaching a circle at all, despite of how optically convincing it might be.
Um, yes...

What's the "nah" referring to?
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PizzaPlanet

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Re: Proof that pi=4
« Reply #19 on: March 16, 2011, 01:52:42 AM »
Englsh's post. You squeezed yours in between the two. Time to edit a quote in, I guess.
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Parsifal

Re: Proof that pi=4
« Reply #20 on: March 16, 2011, 03:07:05 AM »
Quote from: Thork on March 13, 2011, 02:26:26 PM
Actually he is doing what I said (area is directly proportional to dimensions of a shape), but I will make it simpler for you.

Imagine having a piece of string 3.14 blah blah metres long. And you fuse it together at the ends and put it in a circle so the diameter is 1m. Now try pulling the first corner of the square out of it, to make the square. You can't because you need a longer piece of string. What you have done to make it fit the 3.14 circle as you tend towards infinity is infinitely fold the perimeter back on itself so it follows the 3.14 metre outline, but you have enough folds to make it still be 4m long.
He is not squaring the circle. You will find that he is doing the very opposite.
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Thevoiceofreason

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Re: Proof that pi=4
« Reply #21 on: March 16, 2011, 04:16:16 AM »
one thing I find disturbing, is that if there exists a certain quantification of space, i.e. a pixelated world with one length of the pixel being the plank length, might the physical distance be 4?
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Thevoiceofreason

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Re: Proof that pi=4
« Reply #22 on: March 16, 2011, 04:27:05 AM »
Quote from:  Parsifal on March 16, 2011, 03:07:05 AM
Quote from: Thork on March 13, 2011, 02:26:26 PM
Actually he is doing what I said (area is directly proportional to dimensions of a shape), but I will make it simpler for you.

Imagine having a piece of string 3.14 blah blah metres long. And you fuse it together at the ends and put it in a circle so the diameter is 1m. Now try pulling the first corner of the square out of it, to make the square. You can't because you need a longer piece of string. What you have done to make it fit the 3.14 circle as you tend towards infinity is infinitely fold the perimeter back on itself so it follows the 3.14 metre outline, but you have enough folds to make it still be 4m long.
He is not squaring the circle. You will find that he is doing the very opposite.
Motherf*ckn miracles. How does this alt have the same name?
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Saddam Hussein

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Re: Proof that pi=4
« Reply #23 on: March 16, 2011, 04:33:50 AM »
Quote from: Thevoiceofreason on March 16, 2011, 04:27:05 AM
Quote from:  Parsifal on March 16, 2011, 03:07:05 AM
Quote from: Thork on March 13, 2011, 02:26:26 PM
Actually he is doing what I said (area is directly proportional to dimensions of a shape), but I will make it simpler for you.

Imagine having a piece of string 3.14 blah blah metres long. And you fuse it together at the ends and put it in a circle so the diameter is 1m. Now try pulling the first corner of the square out of it, to make the square. You can't because you need a longer piece of string. What you have done to make it fit the 3.14 circle as you tend towards infinity is infinitely fold the perimeter back on itself so it follows the 3.14 metre outline, but you have enough folds to make it still be 4m long.
He is not squaring the circle. You will find that he is doing the very opposite.
Motherf*ckn miracles. How does this alt have the same name?

He put a space before "Parsifal", making his name " Parsifal".  How sneaky.
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PizzaPlanet

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Re: Proof that pi=4
« Reply #24 on: March 16, 2011, 09:09:23 AM »
Quote from: Thevoiceofreason on March 16, 2011, 04:16:16 AM
one thing I find disturbing, is that if there exists a certain quantification of space, i.e. a pixelated world with one length of the pixel being the plank length, might the physical distance be 4?
Well, considering the fact we made up pixels and made them to be squares, I would say it's highly unlikely that our creation (much younger than all of existence) is of the same nature as the universe. I don't think we need to worry about everything around us consisting of pixels.
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JoshuaZ

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Re: Proof that pi=4
« Reply #25 on: March 17, 2011, 11:22:02 AM »
The problem is that the limit of a distance is not necessarily the distance of a limit.
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Thevoiceofreason

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Re: Proof that pi=4
« Reply #26 on: March 17, 2011, 11:37:46 AM »
Quote from: PizzaPlanet on March 16, 2011, 09:09:23 AM
Quote from: Thevoiceofreason on March 16, 2011, 04:16:16 AM
one thing I find disturbing, is that if there exists a certain quantification of space, i.e. a pixelated world with one length of the pixel being the plank length, might the physical distance be 4?
Well, considering the fact we made up pixels and made them to be squares, I would say it's highly unlikely that our creation (much younger than all of existence) is of the same nature as the universe. I don't think we need to worry about everything around us consisting of pixels.
well yes, but if this relates to the coastline problem. The smaller your ruler, the longer the coastline appears to be.
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Trekky0623

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Re: Proof that pi=4
« Reply #27 on: March 17, 2011, 12:17:30 PM »
You could make the same argument by measuring the curvature of all the atoms currently on your arm. It doesn't have to be pixels.
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ﮎingulaЯiτy

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Re: Proof that pi=4
« Reply #28 on: March 17, 2011, 01:29:35 PM »
Presume this image represents fields defined by vectors, not just an arrangement of pixels. Are the black squares touching?

[/douchebaggery]
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JoshuaZ

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Re: Proof that pi=4
« Reply #29 on: March 17, 2011, 01:45:53 PM »
Quote from: ?ingula?i?y on March 17, 2011, 01:29:35 PM
Presume this image represents fields defined by vectors, not just an arrangement of pixels. Are the black squares touching?


That depends on what your precise notion of touching is. I don't know why you want these to be fields and I don't see what you get out of that. If you mean that you want to represent the squares as squares in the abstract not bunches of pixelsl you could just say that. In this context you might need specific descriptions of what is happening at the boundary points to answer this question.
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