Four-Colour Ant (LLRR)
The same machine with four colours — and no highway
The same ant with four colours and the turn sequence left-left-right-right. It never escapes into a highway; instead it keeps circling back and slowly grows a roughly symmetric patch.
How it works
In plain English, before the notation
This is Langton’s ant with four colours instead of two, and a turn instruction for each: left, left, right, right. Changing the turn string changes the outcome completely. Where the two-colour ant escapes into a straight road after about ten thousand steps, this one does not — it keeps folding back over its own trail and grows a compact, roughly symmetric patch that expands slowly and steadily.
Compare turn strings
- Open Presets and load "Four-Colour Ant" and press Play. Let it run to 20,000 steps.
- Note how compact the patch stays and how the ant keeps passing back near its starting point.
- Now switch to Langton’s Ant and run the same number of steps. By 20,000 that ant is far off-screen, building a road.
Starting configurations
Loads straight into the simulatorTry any rule
The catalogue covers a few dozen rules. Here you can run any of the 262,144 two-state grid rules, or any of the 256 one-dimensional rules, including ones nobody has written up.
Well-known rules
Where it came from
Multi-colour ants were introduced as a generalisation of Langton’s: give the machine n colours and a turn string of n letters, and there are 2ⁿ ants to examine. Gale, Propp, Sutherland and Troubetzkoy surveyed them and found that the turn string, not the number of colours, decides the qualitative outcome. Some strings give highways, some give symmetric growth, some give something in between.
The rule, precisely
What each cell looks at
None. The ant reads only the cell it occupies.
What a cell can be
Four colours per cell, plus the ant’s position and heading
The update
heading' = heading turned by turnString[colour]; colour' = (colour + 1) mod 4; position' = position + one step in heading'
The turn string LLRR gives one instruction per colour, read in order: colour 0 turns Left, colour 1 Left, colour 2 Right, colour 3 Right. Langton’s ant is the two-letter string RL.
Bounded, symmetric, slow
The behaviour is qualitatively different from the two-colour ant, and consistently so:
- The painted region stays compact and grows slowly rather than escaping in one direction.
- The pattern is close to symmetric about the starting point, which is characteristic of ants whose turn strings are balanced between left and right.
- The head repeatedly returns to the neighbourhood of where it started instead of drifting away.
- No highway phase appears, at any step count that has been checked here.
Unknown for this particular turn string
Universality has been established for the two-colour ant. Whether it carries over to a specific multi-colour turn string is a separate question and has not been settled for LLRR.
Sensitivity to the rule, not the initial state
Chaotic systems are usually described as sensitive to their starting conditions. What this pair of ants shows is a different kind of sensitivity: identical starting conditions and a two-letter change to the rule produce entirely different long-term behaviour.
Things to try
- Run both ants at the same speed for the same number of steps. The comparison makes the point far better than either one alone.
- Use the Place Ant tool to add ants with different headings and see whether several LLRR ants together behave differently from one.
- Pause and paint some terrain with the Pencil before starting — the ant’s route through pre-coloured ground is worth watching.
Frequently Asked Questions
References
The turn-string notation and a survey of which strings produce which behaviour.
The general class these ants belong to, and how it relates to Turing machines.