a quick explanation of the collatz conjecture. first, take any positive number (integer) and apply this rule:
- if it’s odd: multiply by 3, then add 1 (e.g. 5 becomes 16)
- if it’s even: divide by 2 (e.g 16 becomes 8)
then apply the rule again to your new number. then again. and again… the collatz conjecture says that if you continue applying this rule you’ll always eventually get to the number 1. this, however, has never been proven, although no number has ever been found to disprove the conjecture.
we can visualize the sequence of numbers with a directed graph like this one to the left. by working backwards from the number 1 we can determine all the possible paths that lead to 1.
by creating directed graphs of 10,000+ numbers and applying certain visualization rules, the results can be pretty stunning.
in this example (from the math themed coloring book where i first learned about the collatz conjecture) each sequence of numbers leading to 1 bends slightly left if it’s coming from from an odd number (n*3 + 1) or bends right if it’s coming from an even number (n/2).
i was immediately struck by how such a simple, rigid rule could create something that looked so organic and chaotic.
on stack exchange and reddit i discovered code that others had written to visualize the collatz conjecture. i modified it in mathematica to achieve the look i was going for.
i ran into difficulty exporting high-res images from mathematica that didn’t render with these little blocks. i messaged Patrick Scheibe (@halirutan) and he generously helped me update the code, as well as suggested i export a black svg and then colorize in illustrator.
the black svg file exported from mathematica.
by grouping paths together in illustrator and applying a ‘within stroke’ gradient to the lines, i was able to get an attractive look that didn’t crash my computer.
i had echod graphics print the resulting image on white acrylic.
i covered the back with white vinyl to make the acrylic less transparent and then attached a wood frame with vhb tape.
bonus: the collatz conjecture appears to have a lot of love in the world of math nerds (xkcd even wrote a comic about it). these are a few other visualizations.