Langton's ant

All squares on the plane are colored either black or white. We distinguish one cell on the plane, and call it the "ant". The ant has a direction of motion in one of the four cardinal directions, and a starting point and direction on the plane. As the ant moves, it changes the color of the squares it moves over.

The rules are simple, and depend only on the color of the square the ant is on at each move:

 Color of square    New color of square    Motion                                 

 black              white                  turn 90° right, move forward one unit  
 white              black                  turn 90° left, move forward one unit

These simple rules lead to surprisingly complex behavior.

Langton's ant can also be described as a cellular automaton, using a range of colors for the "ant" square to encode the black/white state and the direction of motion of the ant.

Extension to Langton's ant

There is a simple extension to Langton's ant where instead of just two colours, more colours are used. The colours are modified in a cyclic fashion. There is also a simple name giving scheme for such ants: for each of the successive colours, a letter 'L' or 'R' is used to indicate whether a left or right turn should be taken. Langton's ant would get the name 'RL' in this name giving scheme.

Some of these extended Langton's ants produce patterns that become symmetric over and over again. One of the simplest examples is the ant 'RLLR'. One sufficient condition for this to happen is that the ant's name, seen as a cyclic list, consists of consecutive pairs of identical letters 'LL' or 'RR' (the words "seen as a cyclic list" tell the last letter may pair with the first one.)

External links

• Further Travels with my Ant (http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims95-1) by D. Gale, J. Propp, S. Sutherland, and S. Troubetzkoy: an article in PostScript or TeX format that contains a proof of the above sufficient condition for symmetric patterns.
• http://www.theory.org/software/ant/
• http://www.complex.iastate.edu/information/download/Trend/examples/langton_ant
• http://home.planet.nl/~faase009/D0107#21
• http://www.hut.fi/~jblomqvi/langton/index Java applet with multiple colours and programmable ants.