## Fractals from iterating sines

In a recent blog-post, John Cook, considered when series such as the following converged for a given complex number z

z_{1} = sin(z)

z_{2} = sin(sin(z))

z_{3} = sin(sin(sin(z)))

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.

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.

LaunchKernels[4]; (*Set up for 4 core parallel compute*) ParallelEvaluate[SetSystemOptions["CatchMachineUnderflow" -> False]]; convTest[z_, tol_, max_] := Module[{list}, list = Quiet[ NestWhileList[Sin[#] &, z, (Abs[#1 - #2] > tol &), 2, max]]; If[ Length[list] < max && NumericQ[list[[-1]]] , 1, 0] ] step = 0.005; extent = 8; AbsoluteTiming[ data = ParallelMap[convTest[#, 10*10^-4, 1000] &, Table[x + I y, {y, -extent, extent, step}, {x, -extent, extent, step}] , {2}];] ArrayPlot[data]

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 on Wolfram’s MathWorld site. Still, it was a fun discovery while it lasted

**Other WalkingRandomly posts like this one:**

How much of this depends on the function being sine and how much of the first n terms of the power series, for n =1, 2, 3, …?

Very nice image. I discussed the iterated sine on my blog here

http://observations.rene-grothmann.de/a-chellenge/

http://observations.rene-grothmann.de/the-iterated-sine/

It is always amazing, how simple things can lead to mathematical beauty.