Pi is 4 video
I mentioned the “proof” of π=4 in a previous post. In time for Pi Day on Monday, I’ve created a new video setting out the problem.
For those not inclined to view videos here is the problem:
Take a circle of diameter 1. Its circumference is π since its radius is 1/2 and a circumference is 2π times radius.
Now put a square round it.
The length of the perimeter of the square is 4 since each side has length equal to the diameter of the circle. Now fold in the corners like so.
Since there was no stretching or shrinking, the length of this new curve is also 4.
Do the process again, i.e., fold in all the corners.
The length of this curve is still 4 since no stretching or shrinking was involved.
Do it again.
The length is again 4.
We can take the limit of this process. The limit is a circle.
Since the jagged curve gets closer and closer to the circle and always has length 4 we can see that the perimeter of the circle has length 4. But the perimeter length is also equal to π.
Therefore, π is 4.
Where is the mistake?
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Correct me please
the (limit) is not a circle
after the first folding step we get two shapes around each quarter of the circle , each shape looks like a triangle with a curved base , let me call this shape a triangle for simplicity , we need to have those triangles to keep the length FOUR , so we can not totally vanish them with more folding
i believe we need to have a maximum number of those triangles to keep it 4 , this number could be calculated by limits or something (still beyond my skills) .
in other words , if the limit is a circle , this means the (triangles) will turn into (points), which means they will lose some length and it won’t be 4.
your response is really appreciated
Dear Akram,
Thanks for your message! I believe that the limit really is a circle. Of course to see this we really need to know what a limit is and that’s tricky to write in a blog post because we can’t use many maths symbols. First though let us label the “jagged” curves, call them C_n for the nth curve. As n gets bigger these curves get closer to a circle. The important point is that if you give me some number E then I can find an N such that all C_n, with n bigger than N, lie within the circle of radius 1+E. That’s what I mean mathematically by limit. We can picture this as the circle of radius 1+E squashing the jagged curve onto the circle as E goes to 0.
If you are happy with the definition of the limit of a sequence, then compare it to this. Maybe that will make it clearer!
Assuming that one can define a metric on ‘shapes’ (which we must be able to do, if the concept of a limit is to make any sense), the mistake is of course assuming that the function which returns the perimeter of a given shape is continuous. If we’re calling the nth curve C_n and the circle C, and the perimeter of C_n is denoted P(C_n), the claim is that the limit of P(C_n) is equal to P(limit of C_n) as n goes to infinity. But this property requires continuity of P, and so all this process really demonstrates is that P is not continuous.
Good answer! Are you an analyst?
I’m a mathematical physicist (with emphasis on the mathematical), but have done quite a bit of functional analysis in the past. I’m pretty sure I have seen this problem before in a slightly different form, possibly trying to show that 1=2.
Indeed there are many versions of this problem, often involving stairs.
Thanks for your solution. Mine does not involve talking about continuity but is along similar lines!
But the jagged curves may approach the perimeter of the circle, but they will never reach it. There will always remain a certain amount of space which is inside the square, but outside the circle, so the jagged edges aren’t a good approximation of the circle’s perimeter. Of course, I may be wrong, as I’m in middle school, but I don’t think that the proof works.
Hi Eshaan,
But I can get as close as I want to the circle. It’s like looking at 1/n as the integer n gets larger and larger. It gets closer and closer to zero but never gets there. Anyhow, the real reason is a bit subtle for middle school (most of university students don’t get it at first!) but it’s good that you are thinking about the answer.
Best wishes,
Kevin
This just tells us that pi<4 as we have calculated just the upper limit
Doesn’t it tell us pi<=4? I mean if we didn't know that pi=3.14..., then we can't yet exclude the possibility that pi=4 from this argument.
This isn’t right because everyone knows pi is equal to 2.71828.
Hey Kevin,
I think taking a “limited” approach ain’t correct…
In caclusian mindset, we take the assumption that we zoom in things end to smooth out.
When we repeat the steps till infinity, It will be a jagged fractal which resembles a circle.
In this case i think has dimension 2,
When we repeat the steps it doesnt become a straight line like in a circle
Is my answer correct if so how do i refine it?