A circle is simply a set of points all the same distance from the same center. From what we know, and this has proven, it is round. However, if you keep taking corners off of the square, and zoom in at some point, it still has corners. This theory does not work because it has corners at whatever iteration you take this to. Yes, the corners may be infinitely small, but by the fact these are infinitely small corners, well, they are still corners. Corners are not equidistant from the same center. So, the shape that these iterations end up with is not a circle, but an infinigon, with an infinite amount of corners; it may look like a circle, but if you zoom in close enough, you will see corners, and it will not be a circle.

assholes, pi is not the perimeter of a circle, but the times the diameter fits on the perimeter. this only proves that for a circle with diameter 1, its perimeter is 4. trolling is one thing, but doing a webcomic with magic happy troll inside stating something stupid is just wrong.

You're retarded.. Pi is the relationship between diameter and circumference as circumference is pi times diameter. If the diameter is 1 then the circumference is 1 times pi which is just pi. The circumference therefore is pi.

You are right, pi is the relationship between the diameter and circumference. The point of this pic, though, is that pi = 4. Watch this yo! C = (pi)d. So solve for pi. pi = C/d. In this particular pic, C = 4 and d = 1. So, using substitution, pi = 4/1. I am not saying you are incorrect, pi does in fact equal pi, just that your argument is flawed. Your argument doesn't prove that you are right, nor does it prove the picture wrong.

Also, the argument that pi is the number of times the diameter wraps around the 'perimeter' is true. However, this pic has a good argument that the number of times the diameter wraps around the figure is four. This seems to be true an innumerable amount of iterations, until, infinity.... What happens there?

"This only proves that for a circle with diameter 1, its perimeter is 4." No it isn't. And no it doesn't.

The flaw in the argument is the assumption that repeatedly removing the corners in this way, approaches a circle. It doesn't. If you use squares (which this image doesn't) I think the resulting shape will approach some kind of octagon.

I've extended the corner cutting, and it turned out that the resulting shape actually approaches the shape of a circle. The octagon theory doesn't work.

An excellent example that every student of calculus should see! Of course, the circumference is not approximated by the sum of lengths of the lines constructed as shown, but by the sum of the hypotenuses of each of the right-angle triangles formed around the edge of the circle (forming a polygon with vertices on the circle).

@ Anonymous posting on November 26, 2010 11:12 PM...

Pi is the ratio of a circle's circumference to it's diameter. Diameter-the width of a circular or cylindrical object Circumference-The perimeter of a circle. Therefore, if his reasoning was right (which whoever posted right after you proved it isn't), then yes, he would've been correct. But as the person who posted after you proved, you are incorrect on every account. =P

The fallacy in this argument can be clearly visualized if we try to derive the perimeter of a circle from the basics.

The formula C = 2(Pi)r is derived by integrating the lengths of small elements along the perimeter. There we assume the infinitesimal element dl = r.db where db is infinitesimal angle subtended at center.

Here if you observe an infinitesimal element, it is not shaped like a straight line - that's what the figure assumes after infinitely removing corners. SO the incorrect result.

It amazes me how many people reject this yet still provide false arguments for why the proof is false or no argument at all! The person who said "The flaw in the argument is the assumption that repeatedly removing the corners in this way, approaches a circle. It doesn't. If you use squares (which this image doesn't) I think the resulting shape will approach some kind of octagon." is correct in this error; however, it obviously is approaching a diamond. A diamond is just a square rotated pi/4 radians, and the circle still being circumscribed in the diamond would imply that the diamond would still have the same volume and perimeter as the square.

I think the problem is different. I'm not sure about it but I might have an explanation why it's wrong simply by comparing it to the Koch's snowflake. The Koch's snowflake is a closed curve that has infinite length yet can be enclosed by a circle of finite radius... It suggests that the length of a curve is determined not only based on dimensions we can apply to the "object" that this curve presents but it also depends on how the border looks in ifinitesimal scale.

You can see that a circle is perfectly smooth even in the infinitesimal scale. The circle approximation using initially a rectangle with succesively cut corners (or fragments that stand out)is not smooth though. The fact that it's not smooth results in some deviation from what we get for a real circle...

PS: I'm also not sure and I don't like saying such things without a proof but I think that the length of that circle is of course pi, length of a "circle" approximated infinitesimaly with the presented method is 4, but still their areas are the same...

Actually, if you were to cut a circle and roll it out, its length would be 2pi(r) which with the unit circle, is 2pi length, which (not so) coincidentally is also the number of radians in a circle, also an angle of 360 degrees has an arclength of the circumfrence of the circle

I say this deduction makes only an error up to a constant. Which should be fine for most physicians, the do constant errors all the time. :)

For anyone else... The argument: "the stair-curve looks similar so the circle will have the same length, i guess" is wrong (never guess :) ).

The length of this stair-approximation will not converge to the length of the circle, even if the area does converge. It simply and obviously does not.

Stated pseudo mathematically: lim_{n to infinity} P(Q_n) not = P(lim_{n to infinity} Q_n) in many cases, were P stands for any Property and Q_n for a sequence of abstract objects.

Stated differently: the length of the outer border of the limit of the square approximation is not the same as the limit of the sequence of lengths of ever finer approximations to the circle.

If you try to approximate something you have to show the approximation does converge to the right thing. You must show it.

Therefore you first must define precisely WHAT the thing is you want to approximate, here the length of a curve. With an "euclidean length" the method does not even work for arbitrary straight lines e.g. for the line from (0,0) to (1,1) (this has length square root 2, not 2!) You could also define the distance between two points (a,b), (c,d) to be the absolute value of (a-c + b-d). You would end up with a different "metric" or "notion of lengths and distances", and with this notion the method in the cartoon would be ok :)

So if you were not sure were the argument wrong: *very often : property(limit to infinity of some objects) not equal to : limit to infinity of property(some objects) *if you want to use an argument like swapping limit and property be very carefull! You have to explicitely give a reason why this should be ok!

The correct way to approximate pi would be to take the diagonals of the "bumps" and add up all the diagonals' lengths and then you get some number under pi.

This is wrong, look at the squares in the diagram, pretty similar shapes and on a streight line. As the lengths tend to infinty the lines will converge on a square, rotated pi/4 from the original one. The convergences is due to a neat property in the ratios between height and lengths of these segments

The problem of this is the way the squares curve,infinetly many of them does not generate the orginal cicle. If they did then pi would actually be 4 due to failure to grasp the basic properties of calculus, seriously guys.

In an infinite world, you do not go up and down pixels to make a diagonal line. Instead you go straight through the square. So, removing squares doesn't work. To find the perimeter you would have to calculate that going through each each single pixel is the square root of 2!

Not matter what, the perimeter of the square will never equal the circle, and thus Pi cannot equal four.

Even if you were to repeat the process with cutting the square's corners until there was no detectable difference (functional or otherwise), pi would still never equal four, because though there's no perceptible difference, there IS a difference.

In short, pi cannot equal four, as the circle's circumference and the square's perimeter are two completely unrelated things, and you can't transfer properties from one to the other.

"D=2(pi)r D=pi"

D is the diameter of a circle, in other words, D = 2r. There's no pi involved. I think that you were thinking of Area, as A = 2(pi)r^2.

If you just consider one quarter of the cirle for simplicity, it is quite easy to see that as you approach infinity, the difference between the area of the quarter-circle and the region bounded by the stairstep lines approaches 0, but the difference between the arc-length of the stairstep line and the circle boundary approaches a constant > 0.

Look up Planck's constant - matter/energy is quantized. A circle is theoretical, there's no perfect circle in nature anywhere. That's why engineers and theoretical mathematicians hate each other - they're from different worlds. The universe is like minecraft.

Actually, if you repeat the process infinitely many times, the best you can get is an octagon with each face length being the (square root of 2) - 1. That perimeter is approximately 3.3171. The key to remember is, cut the corners all you want, they will never bend around the circle!

This isn't correct. It LOOKS like a circle, but doesn't become a circle. It has infinite sides. A circle had 1 (or none if a curved line isn't a side).

Nay, a circle is not a polygon, so the perimeter of that circle will be less than the pixelated sides of the square. Archimedes used this logic to say that pi is greater than 3 10/71 and less than 3 10/70.

This is why analysis was invented. Although each approximation A(n) = Circle + E(n) where E is the error, E(n) does not tend to zero as n -> infinity. The L1 norm or Manhattan distance is not equal to the L2 norm or Euclidean distance. The entire premise is false ;-]

How?

ReplyDeleteA circle is simply a set of points all the same distance from the same center. From what we know, and this has proven, it is round. However, if you keep taking corners off of the square, and zoom in at some point, it still has corners.

DeleteThis theory does not work because it has corners at whatever iteration you take this to. Yes, the corners may be infinitely small, but by the fact these are infinitely small corners, well, they are still corners. Corners are not equidistant from the same center. So, the shape that these iterations end up with is not a circle, but an infinigon, with an infinite amount of corners; it may look like a circle, but if you zoom in close enough, you will see corners, and it will not be a circle.

pi=4 only in a pixelated world!!!

ReplyDeletebut infinite resolution pixelation will give a c ircle, i can't find a flaw WTF

ReplyDeletethe pixelated version will always be a zigzag line, and will always be outside of the circle. so, always longer.

ReplyDeleteassholes, pi is not the perimeter of a circle, but the times the diameter fits on the perimeter. this only proves that for a circle with diameter 1, its perimeter is 4. trolling is one thing, but doing a webcomic with magic happy troll inside stating something stupid is just wrong.

ReplyDeleteDude y don't u use ur pen and paper and actually work out the problem for different values of r? This shit actually works!

DeleteYou're retarded.. Pi is the relationship between diameter and circumference as circumference is pi times diameter. If the diameter is 1 then the circumference is 1 times pi which is just pi. The circumference therefore is pi.

DeleteYou are right, pi is the relationship between the diameter and circumference. The point of this pic, though, is that pi = 4. Watch this yo! C = (pi)d. So solve for pi. pi = C/d. In this particular pic, C = 4 and d = 1. So, using substitution, pi = 4/1. I am not saying you are incorrect, pi does in fact equal pi, just that your argument is flawed. Your argument doesn't prove that you are right, nor does it prove the picture wrong.

DeleteAlso, the argument that pi is the number of times the diameter wraps around the 'perimeter' is true. However, this pic has a good argument that the number of times the diameter wraps around the figure is four. This seems to be true an innumerable amount of iterations, until, infinity.... What happens there?

What happens here is the difference between Rationals and Reals

Delete"This only proves that for a circle with diameter 1, its perimeter is 4."

ReplyDeleteNo it isn't. And no it doesn't.

The flaw in the argument is the assumption that repeatedly removing the corners in this way, approaches a circle.

It doesn't.

If you use squares (which this image doesn't) I think the resulting shape will approach some kind of octagon.

I've extended the corner cutting, and it turned out that the resulting shape actually approaches the shape of a circle. The octagon theory doesn't work.

Deletewhen iterated to infinity, the shape would have to approach that of a circle since each iteration uses the surface of the circle as a reference...

Deletecircumference = 2 (pi) r

ReplyDeleteradius = .5

so circumference of this circle is pi. therefore pi= pi

You are both right in a way.

ReplyDeleteThe comic really has nothing to do with anything other than trolling.

and btw

perimeter is P=2(pi)r

Chill dudes. This is just a comic.

ReplyDeleteAn excellent example that every student of calculus should see! Of course, the circumference is not approximated by the sum of lengths of the lines constructed as shown, but by the sum of the hypotenuses of each of the right-angle triangles formed around the edge of the circle (forming a polygon with vertices on the circle).

ReplyDelete@ Anonymous posting on November 26, 2010 11:12 PM...

ReplyDeletePi is the ratio of a circle's circumference to it's diameter.

Diameter-the width of a circular or cylindrical object

Circumference-The perimeter of a circle.

Therefore, if his reasoning was right (which whoever posted right after you proved it isn't), then yes, he would've been correct. But as the person who posted after you proved, you are incorrect on every account. =P

What! I just memorized the first 500 digits for nothing! Pi is only four. :(

ReplyDeletedude, this is just a trick and fun thing.

DeleteThe fallacy in this argument can be clearly visualized if we try to derive the perimeter of a circle from the basics.

ReplyDeleteThe formula C = 2(Pi)r is derived by integrating the lengths of small elements along the perimeter. There we assume the infinitesimal element dl = r.db where db is infinitesimal angle subtended at center.

Here if you observe an infinitesimal element, it is not shaped like a straight line - that's what the figure assumes after infinitely removing corners. SO the incorrect result.

Pi is still 3.141....

It amazes me how many people reject this yet still provide false arguments for why the proof is false or no argument at all!

ReplyDeleteThe person who said "The flaw in the argument is the assumption that repeatedly removing the corners in this way, approaches a circle.

It doesn't.

If you use squares (which this image doesn't) I think the resulting shape will approach some kind of octagon." is correct in this error; however, it obviously is approaching a diamond. A diamond is just a square rotated pi/4 radians, and the circle still being circumscribed in the diamond would imply that the diamond would still have the same volume and perimeter as the square.

It doesn't approach the shape of a diamond. I've extended the corner cutting in paint and determined the result shape does approach that of a circle.

Delete(Correction to above post: area, not volume)

ReplyDeleteMaybe people should be less up their own arse and just enjoy a good bit of mathematical banter?

ReplyDeleteSuccess OP

ReplyDeleteI WILL SHOOT YOU IN THE FACE, YOU JUST UNRAVELED THE FABRIC OF SPACETIME!

ReplyDeleteI think the problem is different. I'm not sure about it but I might have an explanation why it's wrong simply by comparing it to the Koch's snowflake. The Koch's snowflake is a closed curve that has infinite length yet can be enclosed by a circle of finite radius... It suggests that the length of a curve is determined not only based on dimensions we can apply to the "object" that this curve presents but it also depends on how the border looks in ifinitesimal scale.

ReplyDeleteYou can see that a circle is perfectly smooth even in the infinitesimal scale. The circle approximation using initially a rectangle with succesively cut corners (or fragments that stand out)is not smooth though. The fact that it's not smooth results in some deviation from what we get for a real circle...

PS: I'm also not sure and I don't like saying such things without a proof but I think that the length of that circle is of course pi, length of a "circle" approximated infinitesimaly with the presented method is 4, but still their areas are the same...

Actually, if you were to cut a circle and roll it out, its length would be 2pi(r) which with the unit circle, is 2pi length, which (not so) coincidentally is also the number of radians in a circle, also an angle of 360 degrees has an arclength of the circumfrence of the circle

DeleteI say this deduction makes only an error up to a constant. Which should be fine for most physicians, the do constant errors all the time. :)

ReplyDeleteFor anyone else... The argument: "the stair-curve looks similar so the circle will have the same length, i guess" is wrong (never guess :) ).

The length of this stair-approximation will not converge to the length of the circle, even if the area does converge. It simply and obviously does not.

Stated pseudo mathematically: lim_{n to infinity} P(Q_n) not = P(lim_{n to infinity} Q_n) in many cases, were P stands for any Property and Q_n for a sequence of abstract objects.

Stated differently: the length of the outer border of the limit of the square approximation is not the same as the limit of the sequence of lengths of ever finer approximations to the circle.

If you try to approximate something you have to show the approximation does converge to the right thing. You must show it.

Therefore you first must define precisely WHAT the thing is you want to approximate, here the length of a curve.

With an "euclidean length" the method does not even work for arbitrary straight lines e.g. for the line from (0,0) to (1,1) (this has length square root 2, not 2!)

You could also define the distance between two points (a,b), (c,d) to be the absolute value of (a-c + b-d). You would end up with a different "metric" or "notion of lengths and distances", and with this notion the method in the cartoon would be ok :)

So if you were not sure were the argument wrong:

*very often : property(limit to infinity of some objects) not equal to : limit to infinity of property(some objects)

*if you want to use an argument like swapping limit and property be very carefull! You have to explicitely give a reason why this should be ok!

see http://math.stackexchange.com/questions/12906/is-value-of-pi-4

ReplyDeleteThe correct way to approximate pi would be to take the diagonals of the "bumps" and add up all the diagonals' lengths and then you get some number under pi.

ReplyDeleteThis makes me wonder how wrong our integrals may be.

ReplyDeleteThis is wrong, look at the squares in the diagram, pretty similar shapes and on a streight line. As the lengths tend to infinty the lines will converge on a square, rotated pi/4 from the original one. The convergences is due to a neat property in the ratios between height and lengths of these segments

ReplyDeleteThe problem of this is the way the squares curve,infinetly many of them does not generate the orginal cicle. If they did then pi would actually be 4 due to failure to grasp the basic properties of calculus, seriously guys.

ReplyDeleteSo using magnets will not allow me to get to the moon?? Damn you troll science

ReplyDeleteremoving more corners will create a square standing on one corner and not a circle - so it still will have a perimeter of 4!

ReplyDeleteD=2(pi)r

ReplyDeleteD=pi

In an infinite world, you do not go up and down pixels to make a diagonal line. Instead you go straight through the square. So, removing squares doesn't work. To find the perimeter you would have to calculate that going through each each single pixel is the square root of 2!

It's impossible.

ReplyDeleteNot matter what, the perimeter of the square will never equal the circle, and thus Pi cannot equal four.

Even if you were to repeat the process with cutting the square's corners until there was no detectable difference (functional or otherwise), pi would still never equal four, because though there's no perceptible difference, there IS a difference.

In short, pi cannot equal four, as the circle's circumference and the square's perimeter are two completely unrelated things, and you can't transfer properties from one to the other.

"D=2(pi)r

D=pi"

D is the diameter of a circle, in other words, D = 2r. There's no pi involved. I think that you were thinking of Area, as A = 2(pi)r^2.

LOL it seems the troll face has been successful in trolling most of you:P

ReplyDeleteCan anyone prove that the difference between the perimeter of the polygon that is formed and the circumference of the circle is ~0.85... (4-pi)?

ReplyDeleteIf you just consider one quarter of the cirle for simplicity, it is quite easy to see that as you approach infinity, the difference between the area of the quarter-circle and the region bounded by the stairstep lines approaches 0, but the difference between the arc-length of the stairstep line and the circle boundary approaches a constant > 0.

ReplyDeleteLook up Planck's constant - matter/energy is quantized. A circle is theoretical, there's no perfect circle in nature anywhere. That's why engineers and theoretical mathematicians hate each other - they're from different worlds.

ReplyDeleteThe universe is like minecraft.

Actually, if you repeat the process infinitely many times, the best you can get is an octagon with each face length being the (square root of 2) - 1. That perimeter is approximately 3.3171. The key to remember is, cut the corners all you want, they will never bend around the circle!

ReplyDeletethis involves calculus.. see my explanation at:

ReplyDeletehttp://infrastruct.blogspot.com/2012/05/why-pi-is-not-4-and-other-mysteries.html

This isn't correct. It LOOKS like a circle, but doesn't become a circle. It has infinite sides. A circle had 1 (or none if a curved line isn't a side).

ReplyDeleteNay, a circle is not a polygon, so the perimeter of that circle will be less than the pixelated sides of the square. Archimedes used this logic to say that pi is greater than 3 10/71 and less than 3 10/70.

ReplyDeleteThis is why analysis was invented. Although each approximation A(n) = Circle + E(n) where E is the error, E(n) does not tend to zero as n -> infinity. The L1 norm or Manhattan distance is not equal to the L2 norm or Euclidean distance. The entire premise is false ;-]

ReplyDeleteFRACTALS MOTHERFUCKER

ReplyDelete...Fractals...

ReplyDelete