The Tinkerbell Map

Every person who loves math knows its beauty, the poetry found in proofs and the art of a fascinating graph. Indeed, the elegance of math can be considered a sort of aesthetic. Or perhaps aesthetic appeal is derived from that which has foundations in mathematics, like the golden ratio so prevalent in Renaissance art. Explanation notwithstanding, I have encountered many beautiful graphs and maps. My time at the Michigan Math and Science Program has provided me with no small amount of graphs to gaze adoringly at, and their beauty has deepened my appreciation for the subject.

I have recently encountered an intriguing graph of something called the Tinkerbell Map. The map is of a dynamical system (this term seems a bit clinical at first but just means it is a mathematical model to describe how something changes over time) with eventual periodicity (it loops back on itself). It is a strange attractor, meaning the points from which it is pulled depend on initial conditions. The map is created by choosing any x and y coordinates, solving for two more x and y coordinates, and plugging those in to find your next coordinates. This process is repeated ad nauseam, with the result being a beautiful swirling pattern similar to that of Tinkerbell in Disney’s Peter Pan.

The formula:

More on strange attractors:

https://www.dynamicmath.xyz/strange-attractors/