Print Page - Fractal in "Collatz iteration"

I found some fractal-ish pattern in Collatz-like iterations. I think it may be to do with cellular automaton, but I'm not sure.

Collatz Conjecture says that given an integer, if it's even, you divide it by 2, otherwise you multiply it by 3 and plus 1. Repeat the process and you'll eventually get into "4-2-1-4-..." loop.

I converted the Collatz iteration rule into the following form:

Code:

input integer M

if( exists integer n that M == 2n)
	M <- n

else	//exists integer n that M == 2n+1
	n <- (M-1)/2
	M <- 3n+2	//equalivant to M <- 3M+1 then M <- M/2

And here's how I draw the pattern:

1.   Begin with a row of integers ( 1, 2, 3 ... n );
2.   Apply the iteration rule to the whole row;
3.   Repeat the iteration and record the result of each step;
4.   Draw every row with black and white, EVEN NUMs-> White, ODD
NUMs -> Black.
5.   Replace "3n+2" with other formulas to get various patterns.

Here's my Mathematica program to implement the above process.

Code:

(*---------------CELL BEGIN---------------*)

BiIter[nInput_, ExprE_, ExprO_, Var_] :=
 Module[
  {in = nInput},
  If[
   Mod[in, 2] == 0,
   (
    in = in/2;
    in = ExprE /. Var -> in
    ),
   (
    in = (in - 1)/2;
    in = ExprO /. Var -> in
    )
   ];
  in
  ];

CollatzPlot[ExprE_, ExprO_, var_, nMaxNum_, nMaxStep_] := Graphics[
  Raster[Transpose[Table[
     Mod[
      NestList[BiIter[#, ExprE, ExprO, var] &, i, nMaxStep],
      2
      ],
     {i, 0, nMaxNum}
     ]]],
  ImagePadding -> None,
  ImageMargins -> None,
  ImageSize -> 512,
  PlotRangePadding -> None
  ]

(*---------------CELL END---------------*)


(*---------------CELL BEGIN---------------*)

g = CollatzPlot[n, 3 n + 2, n, 512, 128];

Export["G3np2.jpg", g];

Show[g]

(*---------------CELL END---------------*)

I drew some images and I put them in the RAR attachment.
Any idea?

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Fractal Math, Chaos Theory & Research => Other types => Topic started by: Khaotik on October 30, 2011, 06:41:13 AM