Editing Langton's ant (section)

==Extension to multiple colors==

[[Greg Turk]] and [[Jim Propp]] considered a simple extension to Langton's ant where instead of just two colors, more colors are used.<ref>{{cite journal |last1=Gale |first1=D. |last2=Propp |first2=J. |last3=Sutherland |first3=S. |last4=Troubetzkoy |first4=S. |title=Further Travels with My Ant|journal=Mathematical Entertainments Column, Mathematical Intelligencer |year=1995 |volume=17|pages=48–56 |doi=10.1007/BF03024370 |arxiv=math/9501233 |s2cid=123800756 }}</ref> The colors are modified in a cyclic fashion. A simple naming scheme is used: for each of the successive colors, a letter "L" or "R" is used to indicate whether a left or right turn should be taken. Langton's ant has the name "RL" in this naming 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 proof involves [[Truchet tiles]].

<gallery caption="Some example patterns in the multiple-color extension of Langton's ants:">
Image:LangtonsAnt-nColor_RLR_13937.png|RLR: Grows chaotically. It is not known whether this ant ever produces a highway.
Image:LangtonsAnt-nColor_LLRR_123157.png|LLRR: Grows symmetrically.
Image:LangtonsAnt-nColor_LRRRRRLLR_70273.png|LRRRRRLLR: Fills space in a square around itself.
Image:LangtonsAnt-nColor_LLRRRLRLRLLR_36437.png|LLRRRLRLRLLR: Creates a convoluted highway.
Image:LangtonsAnt-nColor_RRLLLRLLLRRR_32734.png|RRLLLRLLLRRR: Creates a filled triangle shape that grows and moves after 15900~ iterations.
Image:CA3061-81k7.png|L<sub>2</sub>NNL<sub>1</sub>L<sub>2</sub>L<sub>1</sub>: Hexagonal grid, grows circularly.
Image:CA174906.png|L<sub>1</sub>L<sub>2</sub>NUL<sub>2</sub>L<sub>1</sub>R<sub>2</sub>: Hexagonal grid, spiral growth.
Image:CA50338 animation.gif|R<sub>1</sub>R<sub>2</sub>NUR<sub>2</sub>R<sub>1</sub>L<sub>2</sub>: Animation.
</gallery>

The hexagonal grid permits up to six different rotations, which are notated here as N (no change), R<sub>1</sub> (60° clockwise), R<sub>2</sub> (120° clockwise), U (180°), L<sub>2</sub> (120° counter-clockwise), L<sub>1</sub> (60° counter-clockwise).