# File:CollatzFractal.png

## 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

## Licensing

 I, the copyright holder of this work, release this work into the public domain. This applies worldwide.In some countries this may not be legally possible; if so:I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
The following pages on Schools Wikipedia link to this image (list may be incomplete):

## The best way to learn

Schools Wikipedia was launched to make learning available to everyone. SOS Children is famous for the love and shelter it brings to lone children, but we also support families in the areas around our Children's Villages, helping those who need us the most. Have you heard about child sponsorship? Visit our web site to find out.