{"id":5078,"date":"2013-08-23T17:51:12","date_gmt":"2013-08-23T16:51:12","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=5078"},"modified":"2013-08-23T17:51:12","modified_gmt":"2013-08-23T16:51:12","slug":"fractals-from-iterating-sines","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=5078","title":{"rendered":"Fractals from iterating sines"},"content":{"rendered":"

In a recent blog-post<\/a>, John Cook, considered when series such as the following converged for a given complex number z<\/p>\n

z1<\/sub> = sin(z)
\nz2<\/sub> = sin(sin(z))
\nz3<\/sub> = sin(sin(sin(z)))<\/p>\n

John’s article discussed a theorem that answered the question for a few special cases and this got me thinking: What would the complete set of solutions look like? Since I was halfway through my commute to work and had nothing better to do, I thought I’d find out.<\/p>\n

The following Mathematica code considers points in the square portion of the complex plane where both real and imaginary parts range from -8 to 8. If the sequence converges for a particular point, I colour it black.<\/p>\n

LaunchKernels[4]; (*Set up for 4 core parallel compute*)\r\nParallelEvaluate[SetSystemOptions[\"CatchMachineUnderflow\" -> False]];\r\nconvTest[z_, tol_, max_] := Module[{list},\r\n  list = Quiet[\r\n    NestWhileList[Sin[#] &, z, (Abs[#1 - #2] > tol &), 2, max]];\r\n  If[\r\n   Length[list] < max && NumericQ[list[[-1]]]\r\n   , 1, 0]\r\n  ]\r\nstep = 0.005;\r\nextent = 8;\r\nAbsoluteTiming[\r\n data = ParallelMap[convTest[#, 10*10^-4, 1000] &,\r\n    Table[x + I y, {y, -extent, extent, step}, {x, -extent, extent,\r\n      step}]\r\n    , {2}];]\r\nArrayPlot[data]<\/pre>\n
 <\/p>\n
\"Sine<\/p>\n
I quickly emailed John to tell him of my discovery but on actually getting to work I discovered that the above fractal is actually very well known. There’s even a colour version<\/a> on Wolfram’s MathWorld site.\u00a0 Still, it was a fun discovery while it lasted<\/p>\n
Other WalkingRandomly posts like this one:<\/strong><\/p>\n