English: The Collatz map can be viewed as the restriction to the integers of the smooth real and complex map

${\displaystyle f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right) (3z 1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}$,

which simplifies to ${\displaystyle {\frac {1}{4}}(2 7z-(2 5z)\cos(\pi z))}$. 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

${\displaystyle f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right) {\frac {1}{2}}(3z 1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}$,

which simplifies to ${\displaystyle {\frac {1}{4}}(1 4z-(1 2z)\cos(\pi z))}$.

Iterating the above optimized map in the complex plane produces the Collatz fractal.