Langton's Ant – Random Squares Experiment

See inside

Click on the green flag.
The ant shrinks to the size of a grid square, ten additional squares are randomly placed within a 100-step square with Langton's Ant starting in the center.
There is no method for predicting how the ant will interact with any of the squares. To find out the long term behavior of the ant, you have the allow the ant to plot its path, step by step.
This project lets you explore for yourself what happens if random square are distributed on the coordinate plane.

Notes and Credits

Even though Langton's Ant is a very simple mathematica system, not much has been proven about its behavior. An excellent resource for Langton's Ant and its significance beyond being just a mathematical curiosity can be found by downloading and reading the following article.

It's well known that the ant, when placed on a blank grid, will wander in what appears to be a chaotic manner and then, after 10,000 iterations, begin to build a diagonal pattern called a highway.
The question one can the ask is what happens if a finite number of squares are randomly placed on the coordinate plane. X.P.Kong and E.G.D.Cohen have proven that the trajectory (the path) of the ant is unbounded. That is, the ant in its wanderings is sure to visit grid squares that have not been previously visited. Ian Stewart's article gives a good description of the proof.
For more information about this project visit my blog at

Shared: 27 Dec 2014 Modified: 31 Jul 2017
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