Field notes / Experiment guide
Thousands of rules, one crowded growth peak
How broadly does cube-root-like growth occur among generalised Langton’s ants, and what distinguishes slower and faster cases?
New to these ants? Start here.
An ant moves on a grid of coloured squares. Each colour tells it to turn left (L) or right (R). It advances that square to the next colour, then steps forward. A rule such as RLR is the list of turns, starting at colour zero. On a blank grid the same rule always makes the same trajectory.
Territory means every cell visited so far. Its boundary consists of visited cells with an unvisited neighbour. A highway is a repeating pattern of motion that carries the ant away. A tether means repeated returns to a small home region.
Growing like the cube root of time means taking eight times as many steps to double a linear size. The rotor walker is a comparison model that follows and rotates arrows stored on cells. A seed makes a random starting pattern reproducible; the ant itself can still move deterministically.
We distinguish c, the chance of discovering a cell per boundary step, from kappa (κ), time at the boundary normalized by perimeter divided by area. Explaining their values is harder than using them to account for growth.
A closer look
The idea under test
A census of width, area and boundary statistics should reveal whether compact growers share a common scaling regime.
What we learned
Many compact growers cluster near a width exponent of one third. Boundary-entry and occupation factors organize deviations, but do not close the dynamical explanation.
What this leaves open
These are finite-window measurements with classification and fit exclusions. Same-run growth accounting cannot prove the exponent’s mechanism.
Read the evidence and reproduction recipe ↗
Inside the experiment
Methods, registered predictions, exact commands and saved analysis. These are the working research documents; expand them for the detail behind the introduction.
Methods and original experiment record
probe-growth-census
Migrated legacy investigation (2026-10-14). Its files used to sit in the flat data/ directory (and calib/, occ/, snaps/, dumps*, knock* where listed) with the label prefix below; experiments/_migration/map.tsv gives the exact old -> new path of every file. This stub was generated from file names and script docstrings. It is not the original write-up: the question, timestamped hypotheses, commands and results are in notes/ (mainly notes/frontier-results.md, notes/frontier-status.md, notes/research-frontier*.md; search for the label), and the producing command is usually in the script docstring or a run_*.sh here.
Old labels / prefixes: c, cube, growth, iso, long, pilot, probe, seed, sq
Scripts
scripts/c_letters.py: Item 3 / P2.1: how much of the rule-to-rule variation of c is carried by letters 1,2 vs later ones?scripts/c_model.py: Predict c from a rule's letters using the edge's chirality.scripts/cube_fill.py: Item 28 follow-up: local volume/extent exponents for cubic ants and the rotor3 control, plus V/E^3.scripts/cube_stats.py: Every number the "Why a cube root" section quotes, computed from the probescripts/sq_baseline_window.py: Baseline for item 24: the square-lattice ants' area exponent in the SAME window (k=16..24, area at t=2^k)
Results
17 entries in results/ (0 are .log/.done markers); csv/json/txt outputs, *_stats.txt hold the summaries quoted in the notes.
Notebook
No committed Jupyter notebook is available for this experiment yet.