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Emergence • Ant Nebulae | jvkr •• blog

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Emergence • Ant Nebulae

“Langton's ant is a two-dimensional universal Turing machine with a very simple set of rules but complex emergent behavior,” says Wikipedia. It’s a procedure that can be described as an ant walking on a lattice, which turns left or right depending on the color of the square it has landed on. If it is white, the ant turns right, when black, it turns left. Upon leaving the square the color of that square switches. The sequence below, taken from this Wikipedia page, shows the first 200 steps of the ant in an 11x11 sized lattice.

Animated gif of Langton's Ant

The simple set of rules promises to produce complex emergent behavior. In order to experience such emergent properties I tried to extend this simple model. In its basic form Langton's ant is described by two rules and two states. If the state is 0—white—the rule is R, if the state is 1—black—the rule is L. A shorthand of writing this set of rules is then LR. It would be possible to introduce a third state, eg. gray, in between white and black. The white state then switches to gray, gray to black and black to white again. Three rules can then be applied. They could be written as RRL, or RLL, etc.

At this stage it should be clear that any number of states is possible. The rule set however can also be extended to include a step forward and a turn. The result is that from any square all four neighboring can be reached. An eight state lattice could be navigated using a rule set RRRFRTLL.

A possible method of visualizing the procedure of a Langton’s ant is to see how the lattice changes. A more interesting graphical result is created by keeping track of the amount of visits per square in the lattice and to express that count as brightness. This creates shapes that are reminiscent of star clouds. That is why I named them Ant Nebulae. The picture below shows both the lattice—on the left—and the nebula. The rules applied here are those mentioned above: RRRFRTLL. The image is the result of many thousands of iterations of the program.


Some of the images that can be found across this website are the result of many tens of millions of iterations on a large lattice. The one below is the result of the 28 member rule set TRFTRFRLTRLTLFLLLLLLRFTTLTTL.

Beautiful impression of an Ant Nebula

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