Visualization of the Week: The Collatz conjecture

Jason Davies has created an animated visualization of the Collatz graph in reverse.

The German mathematician Lothar Collatz first proposed what’s now known as the Collatz conjecture in 1937. It says that you can repeatedly take any number n and divide it by two if the number is even, or multiply by three and add one if n is odd, and eventually, no matter what number you started with, the conjecture says you’ll get to 1.

The Collatz conjecture does remain just that — a conjecture. But one approach to proving it is to consider its reverse. In other words, rather than proving that all numbers lead to 1, prove that 1 leads to all natural numbers.

Toward that end, Jason Davies has created a visualization of the Collatz graph in reverse. It shows the orbits of all numbers with a length of 18 or less.

Jason Davies' Collatz graph

Visit his site to watch the graph grow.

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This post is part of an ongoing series exploring visualizations. We’re always looking for leads, so please drop a line if there’s a visualization you think we should know about.

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