Re: Visualization of pi Message #6 Posted by Joe Horn on 16 Aug 2013, 2:35 p.m., in response to message #5 by Jeff O.
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.
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