Image: CollatzFractal

Description: The Collatz map can be viewed as the restriction to the integers of the smooth real and complex map f(z)=12zcos2⁡(π2z)+(3z+1)sin2⁡(π2z){\displaystyle f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+(3z+1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}, which simplifies to 14(2+7z−(2+5z)cos⁡(πz)){\displaystyle {\frac {1}{4}}(2+7z-(2+5z)\cos(\pi z))}. If the standard Collatz map defined above is optimized by replacing the relation 3n + 1 with the common substitute "shortcut" relation (3n + 1)/2, it can be viewed as the restriction to the integers of the smooth real and complex map f(z)=12zcos2⁡(π2z)+12(3z+1)sin2⁡(π2z){\displaystyle f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {1}{2}}(3z+1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}, which simplifies to 14(1+4z−(1+2z)cos⁡(πz)){\displaystyle {\frac {1}{4}}(1+4z-(1+2z)\cos(\pi z))}. Iterating the above optimized map in the complex plane produces the Collatz fractal.
Title: CollatzFractal
Credit: English wikipedia
Author: Pokipsy76
Usage Terms: Public domain
License: Public domain
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