File:CollatzFractal.png
which simplifies to .
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
- ,
which simplifies to .
Iterating the above optimized map in the complex plane produces the Collatz fractal.