File:CollatzFractal.png

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.