       Re: neat sums and pattered randomness

• To: mathgroup at smc.vnet.net
• Subject: [mg52251] Re: neat sums and pattered randomness
• From: Roger Bagula <tftn at earthlink.net>
• Date: Wed, 17 Nov 2004 02:20:22 -0500 (EST)
• References: <cnbn5l\$9vq\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Both pairs based on the projective line and ( Sqrt[n/(1+n)],Sqrt[1/(1+n])
have the property of giving space filling curves, but the
sorted iteration arrays are not very good  lines in several cases.
The Sqrt[] iteration broke my Mod[,1] output
(I got numbers in (x,y) greater than one and diverging)
It appears the powers (2,1/2)  in the sum function
although increasing a kind of randomness,
do not make good b-normal candidate sums.
It isn't clear why the power functions cause this problem
or solution.

(* Pair Iteration   of projective line type *)
(* two sets of constants depend on if 1-n^2 or n^2-1 is used*)
(* no apparent pattern develops in the iteration: a space filling curve
in 2d*)
x[n_]:=x[n]=Mod[x[n-1]*2+If[Mod[n,2]==1,2*n/(n^2+1),(1-n^2)/(n^2+1)],1]
y[n_]:=y[n]=Mod[y[n-1]*2+If[Mod[n,2]==1,(1-n^2)/(n^2+1),2*n/(n^2+1)],1]
x=0;y=0;
a0=Table[{x[n],y[n]},{n,0, 200}];
ListPlot[a0,PlotJoined->True, PlotRange->All]
(* positive version*)
Clear[x,y,a,b,s,g,a0]
x[n_]:=x[n]=Mod[x[n-1]*2+If[Mod[n,2]==1,2*n/(n^2+1),(-1+n^2)/(n^2+1)],1]
y[n_]:=y[n]=Mod[y[n-1]*2+If[Mod[n,2]==1,(-1+n^2)/(n^2+1),2*n/(n^2+1)],1]
x=0;y=0;
a0=Table[{x[n],y[n]},{n,0, 200}];
ListPlot[a0,PlotJoined->True, PlotRange->All]

Respectfully, Roger L. Bagula