Yes you are spot on. Pie's decimal representation is infinite in length, but the number is still finite. If we look at the animation below, we could see that the number is finite. Now, let's recreate that in a smaller scale. Let's choose the smallest Planck's scale as the size of our point - 1.616252(81)×10. Since we are so zoomed in, we can clearly see each point now. In this scale, the number of points in a circle can be now counted so let's create a circle that contains 33 points. Now let the smaller circle represent each point in the circle (we enlarge the points into a circle so that we can see it clearly) Now,... wait, can we postpone our discussion and somebody please tell me what I did wrong in that picture above? Why doesn't it even reach somewhere near the pie constant? You can count the number of points in the circle and the line and it would be the same.
I'll disagree with PI's length being finite, but at this time it is basically unprovable. Last record was at 5 trillion decimal places. Japanese and US whizzes claim news record for pi calculation -- five trillion decimal places
Hmm, I think you've misunderstood me. The number of decimals π has is infinite. But the number itself is finite. If pie is infinite then it should equal to infinity itself. But it isn't, so if we remove what is impossible, however improbable, must be the truth. Therefore, pie is a finite number but the number of its decimals is accepted to be infinite.
I interpret number of decimals as number of digits. How do you interpret it as? We're not talking size of the number, only precision of the number. Ever read Carl Sagan's book "Contact"? The movie didn't have everything the book did, so I won't spoil it here.
Ok here it is. See, in a very very small scale, Pie's number of decimals becomes definite. Or I might have done something wrong... *scratches head
But in reference to this: wikipedia 22/7 exceeds pie. Anyway, any thoughts on what I did wrong? I gave the small circle's diameter to measure Planck's length, since it do not make sense to make it any smaller. Oww, here's the grided view where I came up with the fraction:
There is a standard calculus equation to calculate pi. I'm old enough to remember slide rules, where 3.14 was good enough. I also have been spoiled by calculators, where the norm is 3.1415926 (that from memory). I like the 355/113 (= 3.14159292) myself, which I just found here... pi What is this, the night owl hangout?
I was thinking more along cyclic polygons, the finer you slice it the more accurate you get. Retch pretty well nailed it, you can't use circles unless they are really, really small, then they approach what I was talking about. Polygon circles are made from a series of repetitious short lines forming the boundary of a circle. The more lines, the closer approximation. Unlike your circles they have no thickness to mess things up.
Well, for me (and this is my opinion), a point can be defined as a circle which has a diameter of 0. So in my picture, I simply increased the diameter of the points so we could see it better. I could return the diameter to 0 and it will not change a thing. It'll just make it harder for us to see and count it. I can't see Retch's reply...
Your problem is that you can't approximate 3.14 on your diagram, right? Bill is correct, you need to make your circles smaller. If you make the diameter only from the dots at the center of the current circles, you should get a much better result. Try for half, or quarter circle, if a full one is too tedious. And how is it that your circles fill up exactly the diameter of the big circle? Do you do math or just tweek alot?
Wait! I get it now why it is impossible to reach the number pie. It is because we have applied the Planck's length as the boundary. As you can see in the picture, there can't be anything smaller than Planck's length. Thus, theoretically, a point must have a diameter the same size of Planck's length, otherwise it will not make sense. Think of it as skinning a circle. Skin it equally on all sides until it comes down to the diameter of Planck's length. Now, using that as our base measurement, create a bigger circle with a diameter of 8 planck circles - thus, we will end up with 26 points around its circumference (as shown in the picture above). Since pie is defined as the ratio of the circumference over its diameter we'll get: pie=26/8 or 3.25
But isn't that what we did here: Oh and btw, these mini-circles are the actual points. It just looks like a circle because we zoomed in and enlarged it. The red dot is the midpoint of the diameter. It does not actually exists since there can't be a point smaller than our mini-circle.
Well, I can't make it smaller because anything smaller than the planck's length will not make sense. It's the wall, I've hit the wall where I can't make the smaller circle (or points) smaller.
What is this obsession with Plank's Lenght anyway? And how did you end up with the conclusion that a circle with 8 point in diameter will have 26 in its perimeter. That is wrong and contradicts the whole pi-irrational theory. You can't measure exactly the number of units that fit in the perimeter, for a given diameter. What you can do in order to approximate π better, is to analyze the perimeter in small dots, much smaller than the circles that you drew (say a quarter or a fifth in size) and even then, you should treat them as you did in your second gif. That is, make the diameter start and begin from their centers, not their edges.