Visualization of pi



#15

Wow, here's a fascinating article on the visualization of pi (and some others) that is also beautiful and thought provoking. I never imagined this type of visualization...

The Art of Pi

Thanks,

Bruce


#16

Serendipity... Just discovered the same page! :D

Massimo


#17

Swiss psychologist Carl Jung would call it synchronicity.


#18

You're right Namir.
Serendipity was the way I found that page: not looking for anything related to it. Then I came here and found Bruce's post... Synchronicity indeed!

#19

I'm a big fan of pi and e (phi not so much, it's just (51/2+1)/2, after all), but wouldn't any irrational number (or set of irrational numbers) produce similar, or maybe even visually indistinguishable visualizations?


#20

Quote:
I'm a big fan of pi and e (phi not so much, it's just (51/2+1)/2, after all), but wouldn't any irrational number (or set of irrational numbers) produce similar, or maybe even visually indistinguishable visualizations?

It depends on how you convert the irrational number into integers to plot. If you use merely the digits of the decimal expansion, then I think you're right; when enough are plotted, they all look random. However, if instead of the digits of the decimal expansion, you use the partial quotients of the continued fraction expansion, then surprising patterns often emerge. The three numbers you mentioned are perfect examples.

Using the HP Prime in CAS mode (type exactly as shown):

convert(pi,confrac,'tt') --> [3,7,15,1,292,1,1,1,2...] (no discernible pattern)

convert(exp(1),confrac,'tt') --> [2,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1...] (a pattern!)

convert((sqrt(5)+1)/2,confrac,'tt') --> [1,1,1,1,1,1,1,1,1,1,1,1,...] (the simplest possible pattern of any irrational number!]

-Joe-

Edited: 16 Aug 2013, 2:37 p.m.

#21

And for those who actually want to store those PI images on disk:

PI FileSystem

;-)


#22

I'm fascinated by this. It means every hit song that has ever been written as well as every hit song THAT EVER WILL BE WRITTEN already exists as a sequence in Pi already. Same for great books, mathematical formulae, chemical equations, gene sequences that include the cures for all diseases, etc., etc., etc., .....

So everything we know and will ever know is right there waiting for us. I guess there must be an infinite number of numbers that this is true for. Maybe we would save time in making new discoveries by searching for sequences to try rather than trudging along using the old and tired scientific method. Maybe start by finding and removing everything we already know and then search the remainder.

BTW, as far a storing files is concerned, even breaking a large file into a bunch of smaller segments would still result in a huge compression factor. I like the idea! So, is it possible one could write a specific "random" number generator that would produce any desired sequence of numbers based on a certain seed? That might be another way to store stuff and only need the equation and seed to recreate it. Maybe this is already being done.


#23

Quote:
So, is it possible one could write a specific "random" number generator that would produce any desired sequence of numbers based on a certain seed? That might be another way to store stuff and only need the equation and seed to recreate it. Maybe this is already being done.

There is a lot of work on the use of genetic algorithms for various types of compression. These are somewhat along the lines of what you suggest.

However,entropy theory puts a limit on how much you can compress data losslessly in the general case. So, with regards to your example, either the equation or the seed would need to be large enough that the average compression rate falls within the constraints defined by information theory.

#24

Thanks for the stimulating posting! Thinking is fun!

Quote:
So everything we know and will ever know is right there waiting for us.
Only the finite things. The infinite decimal expansion of pi does not contain within it the infinite decimal expansion of e, no matter how far out in pi you begin. However, this is not really an exception to what you said, because "everything we know and will ever know" doesn't include the infinite decimal expansion of e.

Quote:
Maybe we would save time in making new discoveries by searching for sequences to try rather than trudging along using the old and tired scientific method.
A delightful thought! Unfortunately, the scientific method has a built-in validity filter, whereas pi contains not only every finite truth, but also infinitely many more finite falsehoods, with no method for discerning between the two. Yes, the cure for cancer is encoded in pi, but so are infinitely many more bogus "cures" worded so cleverly that they sound convincing. Therefore the probability of finding the cure for cancer by searching pi for it is effectively zero.

Since our knowledge of the infinite decimal expansion of pi is growing every year, and THAT portion contains a finite number of desired truths, undesired falsehoods, and lots of nonsense, it's very much like the Internet, the finite contents of which can, at any given moment, be represented by a single number... and IS represented by a single number... one gargantuan binary number... from which we extract some truths, some falsehoods, and lots of nonsense, every day.

Disclaimer: The above text exists in pi, an infinite number of times. Therefore any correspondence between it and truth, falsehood, or nonsense, is completely accidental.


#25

Yes, I went off half-cocked on this one. A little more thought revealed all the dilemmas you mentioned--might just as well use a proper random number generator and hope you get lucky. But, as you said, it's been fun thinking about it anyway.

#26

Quote:
It means every hit song that has ever been written as well as every hit song THAT EVER WILL BE WRITTEN already exists as a sequence in Pi already.

While Pi is highly suspected to be a normal number, it's not even known if its digit form a disjunctive sequence. You might use Champernowne constant instead.

Kind regards

Thomas

#27

Buying enough typewriter (or, I suppose, word processors) and training enough monkeys, we could get there too, right? Reminds me of that old science fiction story in which the monkey got close.... From this session interdict
Every fowl of tyrant wing,
Save the eaggle, feather'd kingg
Damn machine the g is sticked.
(R.A. Lafferty)

#28

The "spirograph" pattern emerges from the arrangement of the integers around the circumference of a circle. The outer band includes the paths of all adjacencies. The next band contains all but the adjacencies of order 1, and so forth. The closer to the center, the lower the density because there are fewer paths.

Why are there five bands?


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