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 Author Topic: Could Pi be considered a fractal?  (Read 8398 times) Description: 0 Members and 1 Guest are viewing this topic.
Roquen
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 « Reply #30 on: September 12, 2014, 11:50:24 AM »

Again, in this context all irrational numbers are just as interesting as Pi and most real numbers are irrational.
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hermann
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 « Reply #31 on: September 19, 2014, 09:25:38 PM »

Very interresting lecture from Norman Wildberger on the number $\pi$.

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Chillheimer
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 « Reply #32 on: December 04, 2014, 11:48:59 AM »

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.

.. how can a line be infinately non linear, it's a fractal linearity ?
exactly, it is a very special case of a fractal. like a square is a very special case of a quadrangle

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.
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.
2/
No fractal programs can use a circle to make a fractal, you have mix many iterations to make a fractal from them.
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..
 « Last Edit: December 05, 2014, 09:27:25 AM by Chillheimer » Logged

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youhn
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Shapes only exists in our heads.

 « Reply #33 on: December 04, 2014, 08:10:16 PM »

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 $\frac{Diameter}{Circumference}$ (instead of the inverse). This gives a whole other sequence of digits. Totaly new man! But wait, it's also just $\frac{1}{\Pi}$.

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?
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Chillheimer
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 « Reply #34 on: December 05, 2014, 08:47:04 AM »

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.
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?

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

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?
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..

quoting socrateases earlier post:
It is mathematically sound, but philosophically unacceptable to me.

actually, I just came online to post these 2 pictures I remembered that maybe speak clearer than my clumsy words:
 m-set-bifurcation.jpg (29.83 KB, 733x605 - viewed 388 times.)  fractal_circle_blk_pi_nocc.jpg (36.74 KB, 500x496 - viewed 460 times.) « Last Edit: December 05, 2014, 09:22:38 AM by Chillheimer » Logged

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jehovajah
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 « Reply #35 on: January 20, 2015, 08:40:23 AM »

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$\ sqrt(2)$. 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!

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Chillheimer
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 « Reply #36 on: October 02, 2015, 12:29:30 AM »

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)/
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kram1032
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 « Reply #37 on: October 02, 2015, 03:00:33 PM »

You are taking the interpretation of those videos too far. All they say is that $\pi$ 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 $\pi$ to show up.
 « Last Edit: October 02, 2015, 03:39:26 PM by kram1032 » Logged
quaz0r
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 « Reply #38 on: October 02, 2015, 06:41:27 PM »

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
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kram1032
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 « Reply #39 on: October 02, 2015, 07:11:48 PM »

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Chillheimer
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Just another fractal being floating by..

 « Reply #40 on: October 02, 2015, 08:04:25 PM »

@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:
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kram1032
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 « Reply #41 on: October 02, 2015, 08:36:13 PM »

fair enough, if you are satisfied with this result
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hermann
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 « Reply #42 on: August 13, 2017, 02:17:20 PM »

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

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

Hermann
 « Last Edit: August 13, 2017, 11:38:16 PM by hermann » Logged

vinecius
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 « Reply #43 on: August 13, 2017, 11:41:57 PM »

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

<Quoted Image Removed>

amazing piece of math.  There's probably a plot to be made from that that could provide some visualization of the self similarity
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greentexas
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 « Reply #44 on: August 22, 2017, 04:17:38 AM »

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.
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