Why is this hyperbolic background invariant?
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I have a mathematical question for you, because it is a mystery for me.
Look at this animation:
I get it as follows. Each frame corresponds to a value of t∈[0,3[ (I take 160 values of t subdivising [0,3[). Here is how I get the frame corresponding to one value of t:
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for each point in the unit square S=[0,1]2, I take its complex affix z, and I send z to the open upper half-plane H={z|ℑ(z)>0} with a conformal map ψ from S to H;
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I attribute a color to Rt(λ(ψ(z))) where R is the Möbius transformation of order 3 defined by R(z)=−1z+1 and λ is the modular lambda function.
The modular lambda function and the conformal map ψ are implemented in my R package jacobi.
My question is: why does the hyperbolic tessellation that we can see as the “background” of the animation not move? Why is it invariant?
I observed the same phenomenon for other modular functions, for example the Klein j-invariant function.
The color mapping C is defined with the help of the HSI color space. The color C(z) depends on the phase of z only. Precisely, C(z) is the HSI color with:
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hue ∈[0,360[ given by the phase φ(z)∈[0,2π[ of z converted to degrees;
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saturation given by √(1+sin(w))/2 where w=2πlog(1+φ(z));
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intensity given by (1+cos(w))/2 with w as above.
Why do we get such a result? I really don’t know.
Before leaving you, let me show you a 3D version of this animation that I made with the isocuboids R package:
I hope you like it.
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