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Chillheimer
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« on: August 13, 2014, 03:13:14 PM » |
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So let's zoom onto the border of a circle. The closer we zoom, the more it looks like a straight line. But compared to each last zoom step what you see is still very self similar to the original, just the curvature is a little less obvious. But the curvature will always be there, no matter how deep you zoom. So it will never be a straight line. And isn't that why Pi has infinite decimal places?`we will always have to round up or down the "last" decimal place. and will never find out the real lenght of 'the coastline of england' - or the circle..  And then: The M-Set is full of circles, all the "blobs" attached to the main body. and: http://www.pi314.net/eng/mandelbrot.phpin my opinion, a circle is just a very special kind of fractal. |
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« Last Edit: August 13, 2014, 03:16:19 PM by Chillheimer »
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--- Fractals - add some Chaos to your life and put the world in order. ---
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cKleinhuis
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« Reply #1 on: August 13, 2014, 03:28:21 PM » |
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that the iteration count converges to pi is new to me  pi itself might be fractal, (obviously, but no rule has formulated for it yet) nevertheless, isnt a fractal a thing that provides a border with a broken dimension? which leads to an infinite length on any subsection, in my point of view this does not qualify for a circle, the circle just does not get longer when subdividing, and as you pointed out it converges to a line when zooming incredible deep, so not a fractal |
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divide and conquer - iterate and rule - chaos is No random!
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Chillheimer
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« Reply #2 on: August 13, 2014, 03:46:20 PM » |
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nevertheless, isnt a fractal a thing that provides a border with a broken dimension? which leads to an infinite length on any subsection, in my point of view this does not qualify for a circle, the circle just does not get longer when subdividing, and as you pointed out it converges to a line when zooming incredible deep, so not a fractal hm.. on the one hand I agree. but then, when I think of the koch curve for example. If you change the length and angle of the variations, it still is a fractal curve, right? (see attached picture, a short scetch I tried with http://dood.al/ ) you could make the angle smaller and smaller, making the form appear more and more circle-like. but when exactly do we cross the point, when the shape isn't fractal anymore? isn't that the same problem we face, as when trying to calculate with pi? the smaller the angle, the more decimal places for pi? (I'm aware, that a circle isn'T made up the same way as my example, but this is the only way I can think of to explain what I mean at the moment) edit: maybe I should have taken "isn't a circle also a fractal" as headline. |
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« Last Edit: August 13, 2014, 04:07:52 PM by Chillheimer »
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lkmitch
Fractal Lover
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« Reply #3 on: August 13, 2014, 06:34:08 PM » |
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So let's zoom onto the border of a circle. The closer we zoom, the more it looks like a straight line. But compared to each last zoom step what you see is still very self similar to the original, just the curvature is a little less obvious. But the curvature will always be there, no matter how deep you zoom. So it will never be a straight line. And isn't that why Pi has infinite decimal places?`we will always have to round up or down the "last" decimal place. and will never find out the real lenght of 'the coastline of england' - or the circle.. And then: The M-Set is full of circles, all the "blobs" attached to the main body. and: http://www.pi314.net/eng/mandelbrot.phpin my opinion, a circle is just a very special kind of fractal. There are certainly concepts that are common between pi and fractals, but I don't think pi itself is a fractal. Pi is a number, not a shape or infinite collection of points. And the fact that a circle's curvature never goes away is a characteristic of all non-linear curves, like a parabola (which has nothing to do with pi). That zooming in on a circle eventually is indistinguishable from a line shows how a circle is not a fractal--zooming in on a fractal will never look like a line. That's the point about fractals. Pi having infinite decimal places is an artifact of our decimal place value notation. The fraction 1/3 has infinite decimal places as well, but is not fractal-like. If there is a parallel structure between numbers and fractals, maybe transcendental numbers (like pi and e) are more fractal-like in some fashion. |
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Roquen
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« Reply #4 on: August 13, 2014, 07:57:55 PM » |
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Sockratease
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« Reply #5 on: August 13, 2014, 10:47:59 PM » |
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I doubt pi could ever be considered a fractal. Just because I don't think so. Can't support my assertion any more than that (but it's good enough for me!). I am curious though... Every time I have looked for a formula for pi, I have never found one!! All I get is the standard definition about radius and diameter. But when seeking 2 actual numbers for really doing the calculation, nothing ever shows up. I want 2 numbers I can do a simple division and get an answer that agrees with the given values in text books for at least 100 digits, preferably more. Does anybody have 2 such numbers? The only 2 I ever find given are the ancient approximation of 22/7 which fails miserably in terms of really giving this string of numbers we are all familiar with. I don't understand how this can be so universal, yet so elusive  |
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Roquen
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« Reply #6 on: August 14, 2014, 06:10:31 AM » |
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Sockratease
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« Reply #7 on: August 14, 2014, 10:41:21 AM » |
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Oh yes, I have seen the many formulas involving summations and trigonometry, but I always thought (purely from a Philosophical point of view) that if the definition of pi was "the ratio of a circle's diameter to it's diameter" that it should be derived from a simple division and have many possible choices for the variables. But I only ever see longer formulas involving other operations than just plain division. This never made sense to me. I just thought it should have a more elegant solution more in keeping with the way we define it. Just the philosopher in me balking at math things which I feel don't work "as advertised" I know there are good reasons for this, but it's just that the several times I tried using just derived values for a circumference and diameter, consciously trying to bypass the formulas and get the value from simple numbers without formulas beyond "a/b" they never agreed with book values for very long. I just enjoy poking at the foundations of these sorts of things. |
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Roquen
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« Reply #8 on: August 14, 2014, 11:09:00 AM » |
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You'd have to invent a numbering system...assuming that's possible. It's an irrational number. Distances along the arc and linear distances are different symmetries so it's might be expressible as pure division is some conformal system (lines and circles are the same)...humm.
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laser blaster
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« Reply #11 on: August 14, 2014, 05:55:29 PM » |
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Oh yes, I have seen the many formulas involving summations and trigonometry, but I always thought (purely from a Philosophical point of view) that if the definition of pi was "the ratio of a circle's diameter to it's diameter" that it should be derived from a simple division and have many possible choices for the variables.
But I only ever see longer formulas involving other operations than just plain division.
Have you heard of the continued fraction representations of pi? For example, this one: http://functions.wolfram.com/Constants/Pi/10/0002/. It's a very simple recursive formula that only involves division and the sequence of square numbers. It's also possible to approximate PI using geometrically inspired iterative methods. Or maybe you're looking for rational approximations to PI. 355/113 is accurate up to 8 digits (including 3). Here's an article that explores other rational approximations: http://blog.wolfram.com/2011/06/30/all-rational-approximations-of-pi-are-useless/The general rule is that rational approximations require you to memorize roughly as many digits (including both numerator and denominator) as digits of accuracy they provide in decimal form. However, some approximations perform better or worse than average. 355/113 is one of the standouts. |
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« Last Edit: August 14, 2014, 06:05:19 PM by laser blaster »
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tit_toinou
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« Reply #12 on: August 14, 2014, 07:48:02 PM » |
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I have to precise that what I was talking about is NOT random walks. It's an algorithm in which the direction of the walk is linked to a property of the number and works for others constants than PI. The images reveals the true nature of such complicated numbers. It has been applied to sqrt(n), e, ln(2), catalan number, etc... and produce really interesting fractals patterns. |
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Roquen
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« Reply #13 on: August 14, 2014, 07:51:44 PM » |
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Sockratease
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« Reply #14 on: August 14, 2014, 10:49:43 PM » |
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Have you heard of the continued fraction representations of pi? For example, this one: http://functions.wolfram.com/Constants/Pi/10/0002/. It's a very simple recursive formula that only involves division and the sequence of square numbers. It's also possible to approximate PI using geometrically inspired iterative methods. Or maybe you're looking for rational approximations to PI. 355/113 is accurate up to 8 digits (including 3). Here's an article that explores other rational approximations: http://blog.wolfram.com/2011/06/30/all-rational-approximations-of-pi-are-useless/The general rule is that rational approximations require you to memorize roughly as many digits (including both numerator and denominator) as digits of accuracy they provide in decimal form. However, some approximations perform better or worse than average. 355/113 is one of the standouts. I figured you'd need additional digits to get additional precision, but it still strikes me as excessively weird that a known circumference and a known diameter require such extravagant methods to derive "true" values for pi. Makes me question the whole idea really. It is mathematically sound, but philosophically unacceptable to me. Good thing I don't write the rulebooks for this stuff! |
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