File:CollatzFractal.png

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Summary

Description	English: The Collatz map can be viewed as the restriction to the integers of the smooth real and complex map $f(z)=\frac 1 2 z \cos^2\left(\frac \pi 2 z\right)+(3z+1)\sin^2\left(\frac \pi 2 z\right)$ , which simplifies to $\frac{1}{4}(2 + 7z - (2 + 5z)\cos(\pi z))$ . 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 $f(z)=\frac 1 2 z \cos^2\left(\frac \pi 2 z\right)+\frac 1 2 (3z+1)\sin^2\left(\frac \pi 2 z\right)$ , which simplifies to $\frac{1}{4}(1 + 4z - (1 + 2z)\cos(\pi z))$ . Iterating the above optimized map in the complex plane produces the Collatz fractal.
Date	21 October 2005
Source	English wikipedia
Author	Pokipsy76

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