A striking pattern is a beginning. Change an assumption, make a prediction, and find out what the picture can actually tell you.
FIRST PRINCIPLE
These are exact experiments in a finite model. Their value is learning to separate what we observed, what the rule guarantees, and what needs evidence from elsewhere.
BEFORE THE FIRST RUN
One row is space. The next row is time.
Each cell is either 0 or 1. Its left neighbor, itself, and its right neighbor determine its next state. All cells update together. Eight possible neighborhoods, each with two possible outputs, give 256 elementary rules. The rule number encodes that eight-bit lookup table. [1]
Wrap around
The rightmost and leftmost cells are neighbors. Imagine a ring, cut open only for display.
Fixed zero
Neighbors outside the row always read as zero. Edge cells still update; they are not pinned.
The stored atlas is 256 rules × 8 seeds × 2 boundaries, each at 128 cells and 128 updates. The laboratory creates additional runs in your browser. It preserves the stored collection and makes no claim to exhaust every starting state.
01
RULES 0 & 204 / EXPERIMENTAL CONTROLS
Trust your instruments.
An experiment with an obvious answer can reveal a misleading measurement.
Rule 0 always outputs zero. Rule 204 always copies the center cell. You can establish both facts by reading their eight neighborhood cases, before running a simulation. These are useful controls: one erases a seed after an update, and the other keeps it forever.
PREDICT BEFORE YOU RUN
If Rule 204 preserves a perfectly alternating row, can its occupancy entropy be maximal while its activity is zero?
Set up
128 cells, 128 updates, wrap boundaries. Compare 0 / single against 204 / alternating. Keep the central one-bit perturbation for both.
Measure
Inspect generation 0, generation 1, final density, activity, occupancy entropy, and the difference view. For Rule 204, check whether every row is identical.
Expect
Rule 204 has occupancy entropy 1 bit and activity 0. Its one-bit difference persists: damage is 1/128 at every generation. Rule 0 has final damage 0, though mean damage includes the changed initial row.
Keep the claim narrow. Occupancy entropy counts the balance of zeros and ones. It ignores their arrangement. A high value does not establish randomness or complex evolution.
TAKEAWAY
A metric becomes useful when you know a case that would fool it.
The familiar triangle has a less familiar ending on a small ring.
Rule 90 outputs the XOR of the left and right neighbors: a cell becomes live when exactly one of them is live. The single-cell pattern is a classic example of nested structure. [2] On the atlas’s 128-cell ring, however, every cell becomes zero at generation 64.
PREDICT BEFORE YOU RUN
Will the triangle disappear at the same time if the two edges stop talking to each other?
Set up
Rule 90, single seed, 128 cells, 128 updates. Compare wrap with fixed zero. Move to generation 64. Then repeat at widths 64 and 256.
Measure
Count live cells at generation 64 and inspect exact recurrence. On the 128-cell ring, the empty row first occurs at 64 and repeats at 65. With fixed zero, one live cell remains at 64.
Explain
XOR is addition modulo two. After 64 updates, the two contributing copies of a seed are shifted 64 cells left and right. On a ring of length 128 those positions coincide, so they cancel. The same argument gives an empty row by width/2 for these power-of-two rings.
A useful confound. All three browser widths—64, 128, and 256—are powers of two. Repeating the result at those widths does not test an arbitrary ring. Use the 127/129-cell extension to change that assumption.
TAKEAWAY
Your world’s size and edges are part of the model, not merely the frame around its image.
A difference map turns “these look different” into a precise question.
Rule 30’s irregular single-cell pattern helped motivate the study of simple rules with complicated output. [3] Here, explore a more specific question: how much does one changed starting cell alter this particular finite trajectory?
PREDICT BEFORE YOU RUN
Will a one-cell difference disappear, remain localized, or spread? Will the answer depend on whether the initial row is empty around it?
Set up
Rule 30, 128 cells, 256 updates, wrap. Compare single and balanced seeds. Select the difference view in both panels. Each panel compares its own seed with a copy in which only the center bit is toggled.
Measure
Read damage at generations 0, 16, 32, 64, 128, and 256. Record the maximum you observe and whether it later decreases. Compare the final value with the time average.
Check the control
For the single seed, toggling the center removes its only live cell. The comparison run stays empty under Rule 30. Its damage therefore equals the original run’s live-cell density at every generation. This is a special comparison, not a generic perturbation sample.
Damage is a measurement, not a diagnosis. One location, one seed, and one time window do not establish mathematical chaos, a Lyapunov exponent, cryptographic quality, or fundamental unpredictability. On a ring, differences may eventually wrap around and interact.
TAKEAWAY
Write down exactly what you changed. Even a one-bit intervention can have a very special meaning.
An important theorem starts where visual resemblance stops.
Matthew Cook proved that Rule 110 can support universal computation by constructing an encoding through cyclic tag systems and interacting structures. His construction uses an infinite row with carefully arranged, repeating material around the encoded data. This is a claim about computational capability under suitable conditions. [4]
PREDICT BEFORE YOU RUN
What would you need to observe to distinguish a moving pattern from a signal that encodes and transforms information?
Set up
Rule 110, 128 cells, 512 updates, wrap. Compare single and balanced seeds. Inspect repeated textures and the boundaries between them at several generations.
Measure
Choose one recognizable structure. Track its position and shape over several updates. Ask whether it repeats in place, translates, changes after a collision, or only appears similar at the display’s scale.
Go deeper
Read Cook’s sections 3 and 4. Identify the background, the symbols being encoded, and the collision rules used by the construction. List the pieces that this browser experiment does not implement.
The laboratory is not a universality test. These ordinary finite seeds do not recreate Cook’s construction. Every fixed finite run has only finitely many states and eventually repeats; finding or failing to find a cycle here cannot settle the infinite-system theorem.
TAKEAWAY
Capability, construction, and observation are different kinds of evidence. Name which one you have.
A road needs vehicles to carry flow. It also needs empty space.
Interpret a live cell as one car. Rule 184 moves it one cell right if the next cell is empty; otherwise it stays put. On a ring, this rearranges cars while preserving their total number. The movement interpretation follows directly from the rule’s lookup table. [5]
PREDICT BEFORE YOU RUN
Which road carries more flow: 32, 64, or 96 cars on a 128-cell ring? Why might adding cars eventually lower throughput?
Set up
Use the traffic experiment: 128 cells, 256 updates, wrap, shuffle seed 12648430. Run exact counts of 32, 64, and 96 cars. The initial arrangements use a deterministic shuffle, not the atlas’s approximate sparse/balanced/dense seeds.
Measure
Confirm the count stays constant. Read instantaneous flow and the mean over the final 128 updates. Flow is the fraction of cells containing a car with an empty right neighbor before an update.
Compare with theory
For this deterministic ring, the long-run relation is q∞ = min(ρ, 1 − ρ). [8] Flow is limited by cars below half occupancy and by empty spaces above it. The asymptotic values for these counts are 0.25, 0.50, and 0.25. Check the finite run against that expectation.
In their 1992 freeway model, Nagel and Schreckenberg added multiple speed states and probabilistic slowing, and compared simulated behavior with real traffic observations. Rule 184 captures only the deterministic, one-cell-speed limiting case; this laboratory does not implement their full model. [6]
Keep the units honest. These are dimensionless model flows, not vehicles per hour. There is no calibration to a road, driver reaction time, lane change, or detector dataset. With fixed-zero boundaries, cars can leave at the right and none enter from the left, so the ring’s conservation and flow relation no longer apply.
TAKEAWAY
A minimal model can isolate a constraint without becoming a forecast of the world.
A resemblance can suggest a question. It cannot answer it by itself.
ILLUSTRATION / THE NATURAL WORLD
AI-generated editorial illustration; not a measured specimen. This image is not a cellular automaton output or biological evidence.
Shell pigmentation is deposited as the animal grows. Boettiger, Ermentrout, and Oster proposed a neurosecretory model in which local excitation and lateral inhibition produce a range of patterns, and compared its responses to growth and disruption with shell examples. Their model contains biological assumptions and interactions beyond a binary three-neighbor rule. [7]
PREDICT BEFORE YOU RUN
If two processes make similar stripes or triangles, what additional observation could tell their mechanisms apart?
Set up
Compare Rules 90 and 30, both single seed, 256 cells, 128 updates, fixed zero. Treat these as pattern-generating models. Read the shell paper’s growth and perturbation examples alongside them.
Measure
Choose a feature before looking: repeated spacing, direction of a boundary, or a response to a local interruption. State how you would measure it in each system. Record which units or biological variables the automaton lacks.
Design the next test
Propose a prediction beyond appearance—for example, how a measured pigmentation process responds to an injury or to a wider growth edge. Explain which observations would disagree with the proposed mechanism.
Analogy has a boundary. The illustration above supplies no observations to fit or validate. This site does not simulate a shell’s growth, identify its species, or infer a biological mechanism. Use measured specimens and the paper’s documented evidence for biological claims.
TAKEAWAY
The most productive resemblance leads to a test that could prove your explanation wrong.
For pairs or small groups. One browser per group, paper for predictions, and no coding prerequisite. The learning objective is to justify a conclusion with a controlled comparison and an explicit limit.
00–05
Make one update by hand
Read the eight-bit rule table. Update a short row together and agree that every cell uses the previous row, not partially updated neighbors.
05–12
Calibrate with the controls
Run investigation 01. Each pair explains how entropy can equal one while activity equals zero. Ask for a prediction before showing the measurements.
12–22
Change only the boundary
Run investigation 02. Save the exact two configurations, the generation being compared, and a screenshot or CSV. Discuss why the picture’s edge is an experimental choice.
22–32
Investigate a question
Choose perturbations or traffic. Write a hypothesis, choose one variable to change, record two runs, and look for evidence that would challenge the hypothesis.
32–40
Exchange and reproduce
Give another pair your run link and prediction. They reproduce the result and check whether your conclusion follows from the measured quantity.
40–45
Leave with a bounded claim
Complete: “Under these conditions, we observed ____. This supports ____. It does not establish ____. Our next test would change ____.”
Assessment: look for one accurately stated observation, one controlled comparison, and one meaningful limitation. A surprising image alone is not the learning outcome.
MAKE THE RESULT TRAVEL
A replication checklist.
Record the complete setup. Rule, boundary, width, updates, initial row or named seed, and perturbation index. For traffic, record exact car count and shuffle seed.
State the comparison first. Identify the one variable being changed. If more than one changes, name the confound rather than hiding it.
Keep values with their window. Record generation indices and distinguish final values from averages. A screenshot alone does not preserve the full state history.
Save the run and the data. Copy the laboratory link, export JSON for the full numerical run, and use CSV for generation-level measurements. Keep PNGs as illustrations of the selected view.
Try a disconfirming case. Change a seed, boundary, duration, or size that could reasonably break your explanation. More runs of the same special case are not independence.
Separate provenance from interpretation. Dynamic runs are browser simulations, the 4,096 stored records are generated computational data, and editorial images are AI-generated illustrations.
A CLAIM THAT CAN BE CHECKED
“For Rule 90, a single seed on a 128-cell ring is empty at generation 64.”
It names the rule, starting state, domain, boundary, time, and observable. A reader knows exactly how to challenge it.
PRIMARY READING / FOLLOW THE ARGUMENT
Where these ideas come from.
The investigations and teaching prompts are authored for this project. The sources below support the historical examples and external scientific claims. Exact finite-run statements can be checked against the atlas and laboratory.
The XOR update and nested single-cell pattern of Rule 90. The finite-ring cancellation argument in this guide is an algebraic consequence, not a new discovery.
Primary paper, university-hosted copy. Multiple speeds, random slowing, and comparisons with real traffic; distinct from the deterministic Rule 184 experiment. DOI: 10.1051/jp1:1992277.
