Work on any possible periodicity of the digits of Pi?

[Kyriakos]

Anyone know of some work on this which is deemed as interesting?

[Sphere]

None that I know how. The decimal version of Pi has no periodicity as far as I know.

When it comes to Pi, a lot of people have found infinite series which sum to Pi.

Leibniz Series


Nilakantha Series



Ramanujan Series



(edit: Irrational numbers never repeat or else they would not be irrational. Pi is irrational.)

[Kyriakos]

Thank you

Yes, i also know of some notable infinite series which add up to pi, the most interesting (in my view) and the oldest one being the spiral of Theodoros:



Mentioned already in the mid 5th century BC, in one of the Platonic dialogues (Theaitetos, or On Science).

Basically it is a spiral formed by each new axis being the square root of an integer, so 1, sr2, sr3 etc. Based on triangles due to the hypothenuses being those axis.
Pi appears rapidly as the approximation of the distance between two roundings of the spiral. Iirc Theodoros himself did not notice the pi connection (not sure?) but he devised the spiral to provide fast proof of those square roots of integers which are themselves irrational numbers.

As for periodicity in pi, i am thinking of something in the manner of not the actual digits repeating, but some kind of connections between subsets of them repeating or otherwise forming a pattern.

BTW: don't recall currently (but it is easy to find out, eg by wiki) if pi is approximated in this spiral from both directions (ie nearing/distancing), as Phi is in the Fibonacci spiral. I think it is not, but anyway

[Gaidin]

Anyone know of some work on this which is deemed as interesting?

Irrational Number.

Literally, Pi is just the ratio of a circle's circumference to it's diameter.

[Kyriakos]

"You don't say?"

...

[Gaidin]

I'm just saying. When you title a thread asking about the periodicity of the digits of pi...

Pi is going to have it's definition, and a bunch of attempts at infinite patterns, but no periods.

[Kyriakos]

^The 'non-periodicity' of irrational numbers, you should find out, refers to the actual digits having periodical patterns, or iterations. It says nothing about Functions tied in whatever way to those digits, which can very well have periodicity. I wasn't (obviously) trying to find if an x number of the digits of Pi will appear again after Y positions in the infinite decimals that number has.

Look at it this way: 0,1,1,2,3,5,8,13 etc will never produce the same digits in the rest of the progression. This does not mean the actual progression is not a function with a very specific pattern (in the above case it is the fibonacci series). Likewise one can view digits in a number as an infinite series (or other things, of course, or combinations of many things), and then try to tie that to digits in an number, etc.

[Gaidin]

Yea, but you gotta remember, every one of those progressions is a series. An inconvenient little ... tagged on the end.

[Kyriakos]

Yea, but you gotta remember, every one of those progressions is a series. An inconvenient little ... tagged on the end.

Many series have infinitely many parts, so that is not an issue. Try to ever get to 2 by adding 1 + 1/2 + 1/4 + 1/8 + 1/16 and so on.

[Sphere]

I find the obsession with the decimal form of pie ugly. I know they have competitions and world records for remembering the most digits, people put it on shirts, coffee mugs and even tattoos. But to me it is completely misses the elegance of math.

If you just remember the Leibniz series for pi...



With a simple spreadsheet you can actually calculate out pie to hundreds of digits in a minute or so. If you can remember just this little pattern, you will know the value of pie in a more accurate way than someone who has spent hundreds of hours memorizing the decimal form. (you can also derive the Leibniz series from fairly simple Calculus.)

To me math is beautiful when it makes complex things simple, not the other way around.

[Kyriakos]

^Maybe all complex things can become simple through expansions in math. Seems to be the main point (some claim that 'complexity' itself is mostly a result of errors in examining the problem).

And since we speak of irrational numbers, it is one thing (and pointless in my view too) to try to remember many digits for, let's say, Phi, and another to focus on the elegant and perfect fraction form of it centered on the square root(s) of 5



Although i am always interested in the geometric forms of things, so ultimately the golden spiral in this case, which i tend to think of as an infinite, changeless position of the fibonacci spiral.