It is a relatively recent innovation that complex-number computations can be done in the evalhf environment. When combined with plots:-densityplot, this makes escape-time fractals in the complex domain very easy to plot. This fractal is based on the Collatz problem. This Wikipedia article has a high-resolution picture of this fractal. I've switched the real and imaginary axes and reversed the direction of the real axis purely for asthetic reasons.
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Collatz:= proc(b,a) #Axes switched
local z:= -a+b*I, k; #real part negated
for k to 31 while abs(Im(z)) < 1 do
z:= (1+4*z-(1+2*z)*cos(Pi*z))/4
end do;
k #escape time
end proc:
#Test evalhf'ability:
evalhf(Collatz(0,1));
plotsetup(
jpeg, plotoutput= "C:/Users/Carl/desktop/Collatz.jpg",
plotoptions="height= 1024, width= 1024, quality= 95"
);
CodeTools:-Usage(
plots:-densityplot(
Collatz,
-1..1, # imaginary range
-0.5..4.5, #negative of real range
colorstyle= HUE, grid= [1024, 1024], style= patchnogrid,
labels= [Im,-Re], labelfont= [TIMES, BOLD, 14],
axes= boxed,
caption= cat(" Happy New Year ",
StringTools:-FormatTime("%Y")),
captionfont= [HELVETICA, BOLDOBLIQUE, 18]
)
);
memory used=24.08MiB, alloc change=24.00MiB, cpu time=7.78s, real time=7.79s
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Download Collatz_fractal.mw
