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Gaps in Fractal Land occur whenever
$$x\cos^{-1}x=2\pi\frac{p}{q}$$ for $p$ and $q$ realatively prime integers.At such points $x$, the functions assume the $⌈(q+1)/2⌉$ values $\cos(2\pi/q)$ for $r=0,1,..., \lfloor q/2\rfloor$.

How can I highlight gaps in the fractal land by using Mathematica?

list = Table[Cos[n x ArcCos[x]], {n, 1, 25}];

Plot[list, {x, -1, 1}, PlotStyle -> Thickness[0.001],PlotPoints -> 150, 
AspectRatio -> 1/3,ColorFunction -> Function[{x, y}, Hue[y]], 
Background -> Lighter[Black, 0.1], 
PlotLabel ->Text[Style["Fractal Valley      Gaussian Mountain     Oscillation Land", 
Red,24, Italic, Magnification -> 1.8], Background -> LightBlue]]

enter image description here

Plot[list, {x, -1, 0}, PlotStyle -> Thickness[0.001], 
PlotPoints -> 150, AspectRatio -> 1/3, ColorFunction -> Function[{x, y}, Hue[y]], 
Background -> Lighter[Black, 0.1],PlotLabel -> 
Text[Style["Fractal Valley", Red, 24, Italic, Magnification -> 2]]]

enter image description here

Thank you in advance to any one who may be able to give me some ideas

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  • $\begingroup$ What exactly is your question here? $\endgroup$ – MarcoB Feb 2 '17 at 3:28
  • $\begingroup$ @MarcoB I want to do something like this $\endgroup$ – vito Feb 2 '17 at 8:24

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