[seqfan] Re: Prime and composite walks concatenated

[Paolo Lava]

Dear William,



very nice your graphics. I can tell you that I had your same idea but,
instead of you, I turned 90 degrees left at any prime. Funny!



Best regards



Paolo P. Lava


2010/4/26 William Keith <wjk26 at drexel.edu>

>  Eric Angelini wrote:
> > Hello SeqFans,
> > I guess this is old hat -- but cannot find anything on the
> > web or in my books:
> > http://www.cetteadressecomportecinquantesignes.com/PrimeWalk.htm
>
> A different version of this with simpler rules interested me a while back.
>
> Initialize:
> * Start at (0,0) and call it node one.
> * Take one step.  Call arrival point node 2.
>
> Rule:
> * At node n, turn right (90 degrees) if n is prime, else go straight.
>
> -----
>
> Some of the resulting images, generated by Mathematica, can be found at
>
> http://earl.of.sandwich.net/Photos/Images/PrimeWalks/
>
>
> http://earl.of.sandwich.net/Photos/Images/PrimeWalks/PrimeWalk400(611x611).gif<http://earl.of.sandwich.net/Photos/Images/PrimeWalks/PrimeWalk400%28611x611%29.gif>starts out by going north, from near the upper right corner, and proceeds up
> to n=400.  (The starting point can be identified since the step 1-2-turn is
> the only segment that turns after a segment of half the length that commonly
> divides all other segments.)
>
>
> http://earl.of.sandwich.net/Photos/Images/PrimeWalks/PrimeWalk3080(1899x1173).jpg<http://earl.of.sandwich.net/Photos/Images/PrimeWalks/PrimeWalk3080%281899x1173%29.jpg>starts out near the lower left corner, and begins by going left, so as to
> get as much of the image as possible on the screen.  It goes up to n=3080, a
> point chosen for its noticeable ending point at the upper right, so as to
> make a pleasant desktop image.
>
> I, too, am interested in how much like a random walk this path behaves.
>  How fast does it diffuse from the vicinity of the origin?  (Probably a
> question on the distribution of long prime-free segments in the integers,
> mod 4.)  How densely does it cover the envelope of line segments it covers?
>  (Probably a question on the distribution of arithmetic sequences.)
>
> Cordially,
> William Keith
>
>
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