Proof... = 4

[Cogman]

That was in response to this line from Fexigoon:
"lets assume this is zoomed in 10000000000000000x, you still have a corner that fails to match the geometry of the circle due to the algorithm used, which yields the extra perimeter."

This seems analogous to saying:
lets assume this is zoomed in 10000000000000000x, you still have that .000000.....1 that fails to equate to one.

Honestly though, I don't understand the proof of why that OP proof doesn't work. I know the OP proof doesn't work obviously, but the explanation of why it doesn't work is beyond me who doesn't understand all the technicalities used in the explanation.

So because in the algorithm used, infinity is a countable number, it's not a true circle. Only an uncountable infinite number of corners can make up a true circle.

And that would mean that you're counting up irrational numbers instead of rational numbers.

Why would an irrational count of infinite corners make a true circles and not a rational count?

I'm lost. Can some of you math geniuses explain to me?

Let me put it this way, Imagine the OP was done on a equilateral triangle with sides measuring 1. This square doesn't even have to tightly fit the triangle, it can be any size we want it to be, so long as the triangle fits inside of it.. It may even be possible to do if it is smaller just so long as the perimeter is bigger than that of the triangles, I haven't rigorously proved it however.

Now do the same thing, fold corners until you get a tight fitting, what do you find? Magically the perimeter of the triangle is also the perimeter of the square that encompasses it. Yet, we know from simple math that the real perimeter of a equilateral triangle is 3 * a side. What is happening?

Again, the explanation is simply this, we are creating an infinite number of sides and adding them up. By doing this, we come out with a perimeter that is larger than the actual perimeter.

Think of it like the definite integral. You are taking the sum of an infinite number of infinitely thin chunks, and yet somehow they add up to the area underneath the curve. It is the same concept. Things behave non-intuitively when infinity becomes involved.

Just know that infinite addition of infinitely small things can, in fact, come up with a number.

 

[JTsyo]

Cool.

At least now we know that the perimeter of a circle of d=1 made up of infinitely small rectangles is greater than that of a true circle by 4-pi.

Edit: Assuming that the rectangles lie on the outside of the curve, that is (forgetting the term for this). What if all the rectangles were on the inside of the circle, resulting in a somewhat smaller approximation?

Man I fucking hate proofs.

Then you should get Pi-(4-Pi).

 

[blinblue]

Personally I think cake is better :trollface:

I really enjoy the troll-math and troll-physics, some of them are really funny.

Countable vs. Uncountable infinity is always a cool topic, and its something you can show your friends if they have a bit of mathematical inclination. The proof that there is a "bigger" infinity of real numbers between 0 and 1 then there are all integers is a pretty straightforward (and visual) proof. And it really blows your mind if you try to think about it. Infinity is already crazy enough as it is, then you have to accept there are different orders of it.
Fun stuff.

Also, someone should make a "troll" math that shows that if you reorder the alternating series 1-1/2+1/3-1/4+1/5 ... you can make it add up to a different number. That's another cool demonstration you can show people, even those with minimal math skills can follow it. Guaranteed to blow your mind

 

[KIAman]

The proof fails at step 5. You can never "go" to infinity on anything. You can only calculate the results toward infinity.

This is why so many people fail to understand calculus. They can't seperate out the conceptual vs reality.

OP, let me put it this way. Your proof is absolutely correct. In reality, you can never have a true circle. Whatever medium you use to represent the circle in reality, be it shaped metal, drawing on paper, or pixels on the computer screen, will be made of component parts which eventually show the irregularities that support your proof.

 

[IronWing]

But as you mentioned that the length of the sides of the teeth approaches 0 as the number of teeth approach infinity. If the lengths of the sides of the teeth is 0, then circumference = 0 and thus pi=0.

Nope, the rate that the length of the teeth go to zero corresponds with the rate the the number of teeth go to infinity. So the balance is maintained and the perimeter is always 4.

 

[Fenixgoon]

Tell me how this sounds:

If P is a point in circle C centered at point Q with radius R, then the distance between P and Q must always equal R. Thus, if we can prove that the distance between center Q and any one point, P', on the circle-like shape determined by the proposed algorithm G is NOT equal to R, then we show that the resulting shape is not a circle.

If P' exists in only algorithm G, but not C, then its distance from Q must be greater than R, since algorithm G only gets asymptotically limited to C, but never equal. Thus, G produces output that is not always part of C, so it's not a circle.

a much simpler explanation :thumbsup:

 

[disappoint]

The proof fails at step 5. You can never "go" to infinity on anything. You can only calculate the results toward infinity.

This is why so many people fail to understand calculus. They can't seperate out the conceptual vs reality.

OP, let me put it this way. Your proof is absolutely correct. In reality, you can never have a true circle. Whatever medium you use to represent the circle in reality, be it shaped metal, drawing on paper, or pixels on the computer screen, will be made of component parts which eventually show the irregularities that support your proof.

This was my thought exactly and is the only post in here worth a damn.

The step that fails is "repeat to infinity". You can't actually do that. You can only imagine doing that. It's like Trident imagining he has slept with a supermodel. Has he actually done so? No he only imagined he did. It never really happened, and neither did your "proof".

So what's the point of this anyway? Just pure mental masturbation because it isn't real and has no basis in reality.

OP go ahead and continue mentally masturbating while real engineers get on with building real stuff in the real world. Here's a towel, clean yourself up after you're done.

 

[destrekor]

The "approximation" in OP's proof never becomes a circle.

And that's basically all that needs to be said.

At least, that's all I'd say - but I hate getting deep into the circles of math hell, the second or third circle is scary enough.

Suffice it to say, continue to cut corners, if you had the ability to zoom in, the circle would have rough edges, and thus never become a circle. A mathematical circle, if one could zoom in under ever-increasing levels of magnification, would always look perfectly smooth between any two points chosen.
It would look like stair steps (pixelation) if one simply continued to cut corners repeated to infinity.

 

[IronWing]

You'll never be a math superstar by cutting corners.