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# Pi: a fractal?

ocalhoun
So, I've been thinking about pi lately, and wondering if there could be a pattern to the seemingly random sequence of digits within it. Then I thought of something else that goes on and on without repeating:

So, I thought, could a fractal number go on and on without repeating?

I made a quick program in C++ to find out for sure, and it looks like one can.

The algorithm goes like this:
 Code: a = 0 b = 0 c =0 d =0 e =0 >>repeat from here<< 1: a ++ e ++ output & test for repeat of first "00000" 2: a ++ c ++ e ++ output & test for repeat of first "00000" 3: a ++ b ++ c ++ d ++ e ++ output & test for repeat of first "00000" >>repeat<<

This creates a sequence of numbers that progresses like this:
00000
10001
20102
31213...

When it is all run into one long string, it looks random, but actually is not. The first sequence does show up occasionally, as it also would in a random number, but the numbers following it are never the same as the numbers following the first group of five.

What I was wondering is, could pi also have a pattern like this within it, and if it did, would it be possible to find it?
Bikerman
Err...what has that rather simple program got to do with fractals? It seems to me that you are simple adding, in a deterministic fashion, to the digits of a 5 digit string. That's nothing to do with fractals and certainly nothing to do with pi.

If you want to code a nice fractal then try this:
1 Draw any triangle
2 Select a point at random within the triangle
3 Plot the point
4 Select one of the vertices of the triangle at random
5 Move halfway from where you are, to that vertex
6 goto 3

Now that WILL produce a fractal which should astonish (though it will still be nothing to do with pi which is not fractal).
ocalhoun
True, my program there is too simple, but I was trying to make a string of digits that made a fractal-like pattern. (Perhaps the original version where digits would restart at 0 after 9 instead of 10 after 9 would have been better...) I was having trouble trying to apply a fractal pattern to a sequence of numbers.
Bikerman
No. You have missed the whole point of fractals.
Firstly a fractal is a geometric shape, not just a number series.
Secondly a fractal is self-similar at any magnification (ie you zoom in and see the same patterns repeating).
Thirdly it can be defined by a simple recursive procedure.

You have got number 3 but ignored the first 2.
nilsmo
Perhaps instead of fractal-like (self-repeating) you mean something with a pattern?

According to good ol' Wikipedia, "Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found."
ocalhoun
 nilsmo wrote: no simple pattern

I guess what I mean is that it may be a complex pattern, with successive digits depending on the values of previous digits.
nilsmo
Well it would be extremely amazing if you found one, since no patterns have been found apparently.
ocalhoun
^Yes, actually finding a complex pattern is easier said than done. It may be impossible with today's technology to do so. Computers can only try by trial and error so far, and that could take forever because there are so many possible patterns, and if there is a pattern, it is apparently too complex to be apparent to human minds...
Perhaps if we ever have a true AI, it could do it?
Bikerman
Well, if you want to look for patterns in pi then try :
http://www.angio.net/pi/piquery
falacal
The digits of pi may be taken to obtain a fractal geometric shape. It seems an interesting approaching.
Bikerman
 falacal wrote: The digits of pi may be taken to obtain a fractal geometric shape. It seems an interesting approaching.

No, as already explained in some detail - it is not an interesting approach because it doesn't work.
_AVG_
Isn't pi the sum of the infinite series:
4(1/1-1/3+1/5-1/7+1/9-1/11+1/13-1/15+......)

Isn't this the best approach to compute irrational numbers?

Irrational numbers like square roots can be expressed as continued fractions, right?
leontius
 _AVG_ wrote: Isn't pi the sum of the infinite series: 4(1/1-1/3+1/5-1/7+1/9-1/11+1/13-1/15+......) Isn't this the best approach to compute irrational numbers? Irrational numbers like square roots can be expressed as continued fractions, right?

But it's not a pattern of digits ie. you cannot determine certain next number of digits in pi, knowing the previous ones, using this formula.
kody
Mmmm, pi....
http://www.weebls-stuff.com/wab/pie/

Want pi now...
medesignz
I like pi... everyone loves pi
Dennise
Pi is an irrational number, not a fractal.
Bikerman
 Dennise wrote: Pi is an irrational number, not a fractal.

Very good. Now, prove it!
WhiteGrandWizard
Hello, I am new and wish to join the fray.
Have any of you considered the fact that space and time as well as math are three dimensional?
Fractals would no doubt be three dimensional; pi would have to be graphed three dimensionally to see its potential, and where the intersects are located to establish multidimensional equations.
Fractals would be linked by intersections in correlation to pi graphing.
A string theory cannot be complete without balance.
Anton3
Oh my friends, regardless of whether or not pi is technically fractal, it is likely to be at least fractal-like.

The decimal representation of pi is a transform of the mathematical object 'pi' onto the domain of base-decimal representation. The actual object 'pi' exists beyond any one particular representation. The Mandelbrot set exists in some sense, whether you are using a computer to plot it graphically, or are analyzing it theoretically. It is the same set.

There are many transforms or representations of pi (representing it as decimal digits, binary digits, graphics, infinite series). Some appear very ordered (infinite series). Others appear like noise (ex: decimal digit representation), totally random.

But unexpected phenomena may arise even with 'noise'. For example, check out the Ulam spiral, which reveals tentative patterns in the distribution of prime numbers, when graphed a certain way.

Pi may not be technically fractal, but I commend the original poster for his broad way of thinking about these complex mathematical objects, and support his further researches!
Bikerman
I do not understand your point.
pi is irrational, therefore the decimal representation does not repeat.
I don't see how pi can be fractal-like.....
The Ulam spiral is certainly interesting but it is interesting in its own right - not because it is fractal-like....