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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..  :hmh:

And then: The M-Set is full of circles, all the "blobs" attached to the main body.
and: http://www.pi314.net/eng/mandelbrot.php

in my opinion, a circle is just a very special kind of fractal.

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 ;)

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.

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.

Be careful:  http://experimentalmath.info/blog/2014/06/can-pi-be-trademarked-2/  (WTF???)

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   :banginghead:

There are plenty of formulas for PI.

http://en.wikipedia.org/wiki/Computing_%CF%80

http://bit-player.org/2014/the-pi-man

You want 2 numbers.  How big?

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.

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.

Yes, PI is a fractal, maybe you just don't have the right visualisation.

Digits of PI can generate nice images, the only software I know to do that is CloisterWalk : http://fr.sourceforge.jp/projects/sfnet_cloisterwalk/ .
I think it is created by Benoit Cloitre (others images of others constant on his website) : http://bcmathematics.monsite-orange.fr/ .

I tried to create the same images but I failed (for now)...

Here are a few random walks generated using the digits of pi:

http://spacefilling.blogspot.com/2014_03_01_archive.html (http://spacefilling.blogspot.com/2014_03_01_archive.html)

Have you heard of the continued fraction representations of pi? For example, this one: http://functions.wolfram.com/Constants/Pi/10/0002/ (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/ (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 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.

It depends on what you mean by "random".  http://en.wikipedia.org/wiki/Equidistribution_theorem

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!

I disagree - the digits of Pi are *chaotic* and anything *chaotic* can be converted into a fractal via appropriate manipulation - that does not make the original chaos itself "fractal" - for instance any general chaotic random generator (including a truly "real-world" random generator) can be used to produce say fBm fractals but that does not mean that the random sequence or sequences used can be said to be fractal themselves.
The digits of Pi itself in the raw do not exhibit any self-similarity nor do they even resolve to a strange attractor (at least not as far as we know, one would literally have to check the full infinite digits to be certain).

Also since I've often said "everything is fractal" I should clarify - the above is using the standard definition of a fractal - my own definition is simpler and more generic - a fractal is anything that can be produced using math that also produces fractals - this means effectively *all math* is a subset of fractal math and I seriously believe that is the way math (and indeed effectively all science) should be approached in future.

Any pi=a/b expression is a circular argument.  a and/or b must already contain pi.

There is also this formula, computing any given digit of pi directly without computing previous digits:
http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula
It's limited to hexadecimal, but that's rather convenient for computers and easy to convert to binary.

As was said, it depends on what one means by random.  The figures in my blog posting were created using the digits of pi, not with random numbers.  It's called a random walk only because it appears random.

Oddly enough this just came through on twitter:  http://divisbyzero.com/2008/12/09/a-new-continued-fraction-for-pi

The digits of are random and no pattern has been discovered until know.
So one of the main features of a fractal, self similarity of patters on different scales, is missing.

On the other hand very interesting algorithems have been discovered to calculate

The following approximation algorithem shows a nice pattern:
(http://upload.wikimedia.org/math/c/d/4/cd4c4c10652ed976bd6b88d882c9cf6b.png)

By the way:
The following aproximation can be used for : 21053343141/6701487259

A comparable problem I am personal interested in are prime numbers. Which seems to appear random and patterns are hard to find.
It seems to me that prime numbers are the non periodic part of the natural numbers.
I have produced a lot of tables to visualise the distribution of prime numbers on my home page.
I have used a lot of JavaScript for the calculation of the tables which challenges the numerical skills of a browser.
I propose to use the firefox browser (That the one I used for the development). On windows 8.1 the explorer warns that he might crash but survives.
With other Browsers I have no experience.
http://www.wackerart.de/mathematik/primzahlen.html (http://www.wackerart.de/mathematik/primzahlen.html)
http://www.wackerart.de/mathematik/primzahlen_mountain.html (http://www.wackerart.de/mathematik/primzahlen_mountain.html)
The only pattern i can visualy recognise are in the visualisation as Ulam Spriral.
http://www.wackerart.de/mathematik/distribution_pattern.html (http://www.wackerart.de/mathematik/distribution_pattern.html)
Also Gilbreath's Conjecture produces interesting patterns:
http://www.wackerart.de/mathematik/gilbreaths_conjecture.html (http://www.wackerart.de/mathematik/gilbreaths_conjecture.html)

When someone is interested in prime numbers we may open a further thread?

Hermann

I think PI would be more like a random fractal. C10 would be a better example.

I am very sympathetic to your view Sockratease.
Philosophically Kant used this situation to justify mathematics as a discoursive subject, one revealed to man by a transcendental being, hence pi was declared a transcendental number.

The principle of exhaustion applies to pi, and that is another philosophical principle. Pragmatically Archimedes advised commenurability as the important or relevant outcome.

When you move to the Indian Sages the taste for infinite exhaustive calculations as an end in themselves really gets introduced into the consciousness of mathematicians. Shunya being everything is experienced in these practices.the great circles of the years and epochs reflects their completely different but consistent approach to Shunya founded on the circular and spherical vortices of space. Reducing the count modulo 10 and modulo 9 revealed an incredible fractal structure which they represented spatially by a system of gears or clocks .this great fractal metronome we reduce to the decimal system, and in that system pi has a representation as an endless polynomial.

The physical " polynomial" is a set of circular dials frozen at each sector representing the numerals of pi. This endless Arithmos sits within all possible Arithmoi described in this way, and each one can be geometrically arranged in space in any pattern which may itself be a fractal pattern or an endless tessellation.

In the end the pursuit of numerals obscures the richer Spaciometry from which they may be drawn. The beauty of fractals for me is not the numeric processes used to generate them, but the algorithms used to generate surface plots and colours which depict extensive properties in space.

Kant?

Really?

Wow.

I always admired his Logic and am a bit surprised he took that position.  It seems out of place with what I recall from the rest of his stuff  {but my Philosophy courses were as long ago as my math and programming courses, so my memory may well be at fault!}.

It's always fascinated me that when confronted by concepts like infinity so many otherwise rational thinkers fall back on metaphysics to explain things.

But I guess it's an easy way out and it explains things with no further need for questions or elaborations, so it must be very tempting   :alien:

Pi itself is not the same as the decimal sequence of Pi. Pi itself is not chaotic, nor fractal nor random, it is just a real value.

If pi written in something other than base 10 were just 1.12345678....  we wouldn't think of it as a mysterious or interesting number would we?

That is what Roquen pointed out, if, instead of using a decimal sequence, we write pi as a sequence a.bcdefg... which is twice a + 1/(1/b + 1/(1/c + (1/(1/d + ....)))) then Pi is in fact 1.12345678...

Interesting point Tglad.

Pi in base Pi is 10.

Pi isn't interesting for the sequence of numerals, it is  interesting for the special and extremely useful geometry it represents.

Because of this special nature, it gets much more attention than other irrational numbers.  Personally, I find e gives much more interesting results when plugged into various properties of fractal generators than pi  (I like plugging mathematical constants in various places just to see what happens   O0

I would say, pi isn't interesting *just* for the sequence of numerals...  Quite a bit of effort has be spent computing zillions of digits of pi in base 10, when the value of the number (for practical purposes) has been known for a very long time.

The ratio of a circular perimeter to a rectilineal diameter is given the label
The ratio of a semi circular perimeter to that diameter is /2.
A quarter circular arc to a radius is /2.
The ratio of a circular disc to a square that sits between the radii of a quarter sector is .

is not an Arithmos . It is a logos.

Is a logos fractal? I do not think it can be any other.
Euler used the notion of Equivalence classes to deal with the endless iterations involved with logoi.
We are taught to " reduce" the endless fractal nature of ratios to their simplest form, because all logoi say too much!

I thought the same thing as you about PI, surely they are a fractal!!! I had the same genious moment, it's true! Wow, how can a line be infinately non linear, it's a fractal linearity ? :D

1/
A circle can't have an infinite circumference. Dont all fractals do? It only has infinite non linearity. If non linearity can frame a fractal, then a mobeus, sinus, strange attractors, can also be considered as fractals.

2/
No fractal programs can use a circle to make a fractal, you have mix many iterations to make a fractal from them.

3/
They use Square Root to find PI. In fact they add the square root of an infinite number of numbers. Sqrts are often irrational numbers so adding an infinite number of them is super irrational. Pythagoras was obsessed with triangles and sqrt, so he sqrted a circle, that's probably all he thought about for 20 years until he decided to make the diagram that you posted.

4/
The irrational number of PI is a random theoretical number resulting from artificial "2D" construct of theoretical maths copying a random natural patterns as if they have some infinite perfection to them, which in nature can't exist, in the known universe, PI doesnt exist. The number itself is as infinitely chaotic as a circle is infinitely non-linear.


In theory, perhaps the most likely pattern that you could find in PI is that it could show echoes of central symmetry, that one element of PI is a mirror image of another.

Does the number that results from adding an infinate amount of sqrts together, is it more a wet biscuit than a fractal?

Again, in this context all irrational numbers are just as interesting as Pi and most real numbers are irrational.

Very interresting lecture from Norman Wildberger on the number .

http://www.youtube.com/watch?list=PL5A714C94D40392AB&feature=player_detailpage&v=lcIbCZR0HbU (http://www.youtube.com/watch?list=PL5A714C94D40392AB&feature=player_detailpage&v=lcIbCZR0HbU)

I know it's been a while, but I'm working on a short article regarding this, so I reread this thread.

First of all, I'd like to restate my initial question again and say that I consider a circle a fractal.
As it was pointed out a few times, I too do not regard the sequence of the digits of pi as fractal.


exactly, it is a very special case of a fractal. like a square is a very special case of a quadrangle

No, I wouldn' say that. What about the Cantor-set? It doesn't have infinite length.
But the mathematical process that leads you to the aproximate result, the circumference of a circle in my opinion is a process that is 'fractal'..
keep in mind, I'm not a studied mathematician.
I don't agree. What about the circular bulbs of the m-set? I think this is exactly what happens. the m-set forms circles - of course only approximately, as the calculation with pi does as well..

so much for now - too bad that this all takes so much time- I wish it was my paied fulltime-job to research fractals.. ;)

That seems to be a very weird comparison (the mset bulbs VS pi in the unit circle). Approximations give lots of trouble. Because ... is an approximation of a circle really a circle? I do not think so. A very small angular segment of a very wide circle may look like a line, may approach a line ... but it never is a line. The mset bulbs therefore are not circles, but just bulbs. You might talk about the circles you can project within these bulbs, but that more like the inner circle of a triangle for which all sides are tangential.

Infinity is a nice conceptual thing, which we can do fun things. But sometimes we see infinity in about every corner we look. It's just our projection of the concept. Pi does NOT have any infinite on it's own. Only when you try to combine it with our numbering system, you find that (most probably) you need an infinite sequence of digits. Does this give Pi a infinite property? I think not. Just an effect causes by the incompatible finiteness of both Pi and our numbering system.

Let's consider the ratio (instead of the inverse). This gives a whole other sequence of digits. Totaly new man! But wait, it's also just .

What about the discussion between contineous and discrete? Does approaching contineous things with discrete systems always end up with either an acceptable error or an infinite sequence of discrete stuff?

I understand what you're saying and in strict mathematical (euclidean?) terms you're definitely correct.
But with a not so strictly-mathematical reasoning: couldn't I argue that the calculation of the circumference too is an approximation, always depending on how many digits of pi you use? ;)

damn. you put that one well. makes me think..
we just use the "wrong language"?.. hmmm..

to be honest, combined with the english language, this is already where my mathematic-school knowlege is becoming a little fuzzy.
but I understand where you're going..

and I take the easy route and ignore it for now, sorry..  :embarrass:

quoting socrateases earlier post:

actually, I just came online to post these 2 pictures I remembered that maybe speak clearer than my clumsy words:

Assume that a circle can be a disc, and that disc can be cut into sectors by the radius of the circle striking the arc of the disc.

Empirically we find that this appears to be possible 6 times, give or take a few minor sectors. Pragmatically let us define this process as sectoring the circular disc into " precisely" 6 sectors. For the Euclidesn classicists we have here the intended meaning of artios, that is precisely, not Evenly!

Empirically we might more accurately say approximately, and thus once again reveal the intended meaning of perisos, that is approximately not Oddly!

It is thus a formal act that makes a disc eternally divisible into 6 equal sectors.

Let us call the line equal to the radius displacement in length, and also as straight as the diameter line in which it proceeds from the centre to the perimeter, a chord when it strikes off a sector. In fact any straight line that strikes off any sector of a disc at the perimeter of the disc we may call a chord, (c).

Now , rather than just count the sectors let us relate each sector to a square or circular Monas or unit. Such a unit will be used to count off the Arithmos of a specific disc.

Thus for any disc, the radius chord always cuts the disc into 6 monads

Can I find a commensurable monad that counts off any general sector in comparison with another?

That is to say, given any 2 arbitrary sectors can I find a precise monad that counts them both off in whole counts? This is the classical issue of commenurability, not as we like to say " precision", except and unless we Meehan by that precisely artios.

It seems reasonable to employ the device that the count of monads( circular or square) of the sector is expressible by the count of the "radius monads",r,  multiplied by the count of the "chord monads", c, and then " halved".  Here the radius or chord monad is obtained by placing the square monad on the radius or chord, or equally the diameter of the circular monad on the radius or chord of the disc sector.

As the monads chosen become finer it is clear that the count becomes larger , but also the perisos monads become smaller and we approach commenurability .

Can we achieve it?

No!

The exposition is simple.

Expressing the count as rc/2 we see that for a c(hord) count that is equal to the diameter count, D,  rD is the expected count for the disc.

But if I construct rD as a rectangle it is clear it is too small, coverin only a semi circle. The curved Shunyasutras outside the semi circle, can be brought within revealing 2 petal shaped areas that are not accounted for.

These areas sit within a quarter disc each, so it makes sense to again apply( iterate) the formula rc/2 but this time to use as the chord count   r. This comes from Pythagoras theorem, and is immediately declareable as incommensurabie.

However let us make the observation that our formula is biased toward giving too small a result.in this correction the count is clearly biased toward being too big, containing more of the circle than necessary. it is therefore reasonable to stop the iteration at this point because it is incommensurable, but also to expect a pragmatic level of resolution of the count. This means that whatever monad size we use we can expect the count to be factoriseable between the 2 arbitrary sectors by approximately this amount.

I will leave it for the interested reader to declare the result of this Fractal factorisation .

• My apologies for the Snootiness in the style. This is how we mathematicians are trained to speak or write formally. It reveals an endemic arrogance which actually is not justified or justifiable.

As agent Smith ( the Matrix gatekerper) correctly pointed out we mathematicians are viruses. We enter into the realm of the natural philosopher and completely destroy it by replicating ourselves and our own arrogances!

sorry to bring this old thread up, but finally someone who knows her math explains what I mean in a mathematical way and brings this topic to a closure for me. I finally can confidently answer the starting question with a big yes. :)
http://www.fractalforums.com/mandelbrot-and-julia-set/approximating-pi-with-the-m-set-(numberphile)/

You are taking the interpretation of those videos too far. All they say is that is related to the tangent function which is sampled by this process in a non-uniform way. - If anything, the way it is sampled is exponential. Not the particulars of how this causes to show up.

kram also believes that the top 12% of a building can piledrive the lower 88% into microscopic dust and molten steel at the acceleration of gravity, completely violating newton's third law of motion.  just sayin   :D O0 :D

I'm sorry, what? Keep your conspiracy theories out of this please.

@kram:  from my viewpoint (which still isn't strictly theoretical/mathematical) the numberphile video explains exactly what I tried to find out when I started this thread.
as mentioned a few posts earlier in this thread, I don't think the number pi itsels is "a fractal". but I consider a circle a fractal, a very special fractal, with unique properties. I know most will disagree due to different definitions, but for me this is the case and this video proves it in the way I wasn't able to do.

@quazor:  :over:

fair enough, if you are satisfied with this result :)

The Bailey-Borwein-Plouffe-Formula for calculating Pi:

(http://www.wackerart.de/mathematik/plouffe.gif)

The formula can directly calculate the value of any given hexadecimal digit without calculating the preceding digits.
A main charactaristic of a fractal is selfsimilarity on every scale.

When the sum elements of the Bailey-Borwein-Plouffe-Formula are considerd as self simalare, then Pi has a fractal structure!

With this formula I have calculated the first 100 000 Digits of Pi:
http://www.wackerart.de/mathematik/kreiszahl_pi.html (http://www.wackerart.de/mathematik/kreiszahl_pi.html)

Hermann

amazing piece of math.  There's probably a plot to be made from that that could provide some visualization of the self similarity

I wouldn't expect pi to be a fractal since you can't predict the next digit (at least not in decimal).
But, again, there are fractals we haven't found the formulas for.

This is obviously a fractal number.

0.010111010111111111010111010111111111111111111111111111010111010111111111010111010

because of the 27 ones, nines, threes, and ones. The zeroes are a little like Cantor dust.