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http://www.trollcomics.net/big.jpg?im=0022.jpg
For those who don't want to check the image, the logic is this:
Start with a circle of diameter 1. From the definition of π, the circumference of this circle is πd = π × 1 = π.
Now draw the smallest square around the circle that you can. Each length of this square will have side 1, so the perimeter of the square is 4.
Now take a corner of the square, and cut a square chunk out of it. The area of the square has shrunk, but the perimeter is still 4.
Take another corner out the same way, the area has shrunk again, but the perimeter is still 4.
Now keep cutting corners this way - infinitely - until you wrap the former square perfectly around the circumference of the circle. The area of the former square has shrunk quite a bit, but the perimeter is still 4.
Therefore, π = 4.
Thoughts? ^_^
it's illogic there must be some secret or something 
Hi Indi
The image your link points to illustrates the construction very well, and at the same time the flaw. If you were calculating the area of the circle by approximation, all would be well, but you just have to look at the polygon to see its perimeter is not approaching that of the circle, as the "approximation" to each small arc of the circle is like replacing the hypotenuse of a right-angled triangle by the other two sides - not the same length at all!
You could just as well start with a right angled triangle within a square and use the same technique to "prove" the square has the same perimeter as the right angled triangle by replacing the diagonal by a staircase of successively smaller squares.
Hope this clarifies the situation for you.
Ahh...varifold .....anyone got a radon? Ahh....here's one
| infinisa wrote: | Hi Indi
The image your link points to illustrates the construction very well, and at the same time the flaw. If you were calculating the area of the circle by approximation, all would be well, but you just have to look at the polygon to see its perimeter is not approaching that of the circle, as the "approximation" to each small arc of the circle is like replacing the hypotenuse of a right-angled triangle by the other two sides - not the same length at all!
You could just as well start with a right angled triangle within a square and use the same technique to "prove" the square has the same perimeter as the right angled triangle by replacing the diagonal by a staircase of successively smaller squares.
Hope this clarifies the situation for you. |
Yeah, thanks, the situation was quite clear for me before i posted it. -_- This was a question for beginners to mull over, and try to reason out.
Oh well, let me see if i can find another one now....
And the fallacy (yes, I know you knew this already) is confusing taxicab (or Manhattan) geometry with Euclidean geometry. When you take the continuum limit in Euclidean space, you measure along the diagonal of the infinitesimal squares, not along the edges.
Having said that, taxicab geometry has both very interesting consequences, and is much more useful in some "real world" applications. For example, if you ask Google Maps the distance between addresses, it will give you the taxicab-distance -- that is, the distance along the roads which connect those addresses -- rather than the straight line separation.
Actually If u draw the square , u would be able to see some gaps between the circumference of circle and the corners of squares with actally makes the difference.. 
This method of proof will be applicable for area of plane figures but not for the perimeter. You can't approximate the cutting to infinity, as you are actually replacing arcs of small length by sides of bigger lengths.
If it showed the connection between points as a straight line, rather than a right angle, it would be a better approximation.
a^2 + b^2 = c^2, but a+b<c.
How come the area decreases but the perimeter doesnt decrease?
The perimeter doesnt decrease when u take out one small square but reapeating the process, the perimeter should have shrunk.. Shouldnt it?
| Arrogant wrote: | How come the area decreases but the perimeter doesnt decrease?
The perimeter doesnt decrease when u take out one small square but reapeating the process, the perimeter should have shrunk.. Shouldnt it? |
No. This is what makes "grid geometry" interesting and counter-intuitive (and why you should read the earlier postings). Draw the picture yourself on graph paper, and see what happens each time you remove a corner.
the proble is that the circle is more like
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and not as the image says
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| firstroad wrote: | the proble is that the circle is more like
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and not as the image says
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press "quote to see it...
see why it's not equal to 4
http://www.youtube.com/watch?v=D2xYjiL8yyE
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| Arrogant wrote: | How come the area decreases but the perimeter doesnt decrease?
The perimeter doesnt decrease when u take out one small square but reapeating the process, the perimeter should have shrunk.. Shouldnt it? |
No, there will still be gaps in it after doing this many times.
Yea...you CANNOT escape the fact that a circle always maintains the same proportions relative to it's diameter/radius. So by measuring these, and then taking a piece of string or what not, you can prove with physical evidence that a circle is far closer to 3.1 (which is closer to the actual value of pi) than 4.
Okay with pi when you measure pi you include the ~1mm line on each side of the circle for the diameter... but the centre of the line for the circumfrence... does this change the outcome of pi? Also where do every single 'edges' connect simultaneously to the circle with no gaps? After losing some squares... approximately 0.86 of the original 4
It must be a some logic or some errors there :/
Do you know that a court in Indiana said that, officially, "PI=4" ?
http://en.wikipedia.org/wiki/Indiana_Pi_Bill
LOL.
That was quite entertaining to read, i hope they got rid of that bill by now otherwise they'd have square tires on their cars and stuff.
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