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>*Reply-to*: tftn at earthlink.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]=0;y[0]=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]=0;y[0]=0; a0=Table[{x[n],y[n]},{n,0, 200}]; ListPlot[a0,PlotJoined->True, PlotRange->All] Respectfully, Roger L. Bagula tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn at netscape.net URL : http://home.earthlink.net/~tftn