Langton's ant

Langton's ants were invented in 1986 and declared "equivalent to Turing machines".

Some of these extended Langton's ants produce patterns that become symmetric over and over again.

Langton's Ant is such an example.

As with Langton's ant, turmites perform the following operations each timestep:

Langton's ant is a two-dimensional Turing machine with a very simple set of rules but complicated emergent behavior.

Langton's ant has the name 'RL' in this naming scheme.

Contains the proof that Langton's ant is unbounded.

For Langton's Ant this kind of pseudorandom motion happens for about the next 10,000 steps.

However, if we ask a slightly more general question, we realize that we don't understand Langton's Ant at all.

At that point you could say 'Now we understand why Langton's Ant builds a highway.'

Langton's ant is a well-known type of turmite defined on the cells of a square grid.

"Complexity of Langton's Ant".

In 2000, Gajardo et al. showed a construction that calculates any boolean circuit using the trajectory of a single instance of Langton's ant.

Greg Turk and Jim Propp considered a simple extension to Langton's ant where instead of just two colors, more colors are used.

A further extension of Langton's Ants is to consider multiple states of the Turing machine - as if the ant itself has a color that can change.

Multiple Langton's Ants can co-exist on the 2D plane, and their interactions give rise to complex, higher order automata that collectively build a wide variety of organized structures.

The phase space of Langton's Ant consists of all possible ways to put black and white squares on a grid, not just the ones that the Ant puts there when it follows its rules.

Other initial configurations seem eventually to converge to similar repetitive patterns suggesting that the "highway" is an attractor of Langton's ant, but no one has been able to prove that this is true for all initial configurations.

Langton's ant can also be described as a cellular automaton, where the grid is colored black or white, the "ant" square has one of eight different colors assigned to encode the combination of black/white state and the current direction of motion of the ant.

Entities built within Wireworld universes include Langton's Ant (allowing any Langton's Ant pattern to be built within Wireworld), and the Wireworld computer, a turing-complete computer implemented as a cellular automaton.