Experiment 1
Keep threshold = 2. Change only the comparison pattern to alternating.
Check the prediction
It flips every cell each update while density stays 1/2. A constant average can hide continual microscopic change.
Emergence and coarse graining
An average can hide the next state.
Apply a local update rule, compare microstates with the same density and test whether a coarse variable predicts its own future.
Start with: Binary states and averages. Coarse graining means retaining a summary while discarding some microscopic detail.
01 · Commit to a prediction
What do you expect?
No. A single six-cell block stays at six. A five-cell block plus an isolated cell becomes a five-cell block: five active cells. The same initial average hides different neighborhoods.
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02 · Change an assumption
Predict. Change. Explain.
Before changing a control, say what should move and why. Start with the experiments below. Reset restores the starting model; it preserves your written responses.
Experiment 2
Reset. Change only the threshold to 3. Compare the initially half-active rings.
Check the prediction
The six-cell block loses its two endpoints per step; the five-cell block loses endpoints and its isolated cell disappears. Same initial density still does not specify the next density.
1 to 12 · step 1
Calculated model output. The table gives the same values. Displayed values are rounded; calculations keep full precision.
Read the plotted data as a table
Every microscopic state
1 = active; 0 = inactive. Read each row around a ring.
03 · Connect the mechanism
From the picture to the quantities.
Each cell follows a local rule with no access to the whole ring. Collective patterns follow from those rules and the arrangement. Reducing a ring to its active fraction compresses many microstates into one macrostate. A useful macro-variable need not reproduce every detail, but if you want to predict its next value from its current value alone, hidden arrangements must not change that prediction.
xᵢ(t+1) = 1 if xᵢ₋₁(t)+xᵢ(t)+xᵢ₊₁(t) ≥ θ, else 0
ρ(t) = [Σᵢ xᵢ(t)] / 12
ρ(t) = [Σᵢ xᵢ(t)] / 12
xᵢ is a binary cell state, 0 or 1. Indices wrap around the 12-cell ring. θ is threshold 1, 2 or 3. Every cell updates simultaneously from the previous row. t counts updates, 0 to 12. ρ is active fraction; the graph shows percent. The fixed reference ring starts 111111000000. Pattern choice changes only the comparison ring.
Worked example
Compare 111111000000 with 111110010000 at threshold 2. Each endpoint of the six-cell block has itself plus one active neighbor, so the block stays six. In the comparison ring, the isolated 1 has no active neighbor and vanishes, while the five-cell block remains. Thus both start at 6/12, but next densities are 6/12 and 5/12. No random decisions are involved.
Assumptions and limits
This is an original finite cellular automaton illustration, not a fitted model of organisms, neurons or people. Updating sequentially would change the rule. No general claim that complexity or entropy must increase follows. Coarse graining discards information; it does not automatically yield a closed dynamical law. The information-theory connection concerns deterministic summaries and conditional uncertainty, not a thermodynamic calculation.
The annotated sources distinguish established results from this lesson’s original examples.
04 · Follow the structure
Where else does this apply?
Ecological occupancy
A regional average can hide clusters and isolated occupied patches.
Boundary: Dispersal, habitat and stochastic survival need a biological model, not this binary majority rule.
Image summaries
Two images with the same average brightness can have very different edges.
Boundary: Compression is useful only relative to a task; an average cannot preserve arbitrary spatial structure.
05 · Retrieve without hints
Close the explanation. Try a new case.
Write an answer before opening its feedback. Later, return directly here without rereading above. Recognition, explanation and transfer are separate outcomes. No page action or answer reveal measures mastery.
Recognition and explanation
Reveal reasoning and rubric
No. The alternating ring flips every cell at each majority update while retaining six active cells.
Self-check: Provide two distinct consecutive microstates with the same macrostate.
Calculation
Reveal reasoning and rubric
1100 remains 1100. Each active cell has itself and one active neighbor; each inactive cell has only one active neighbor. Density stays 1/2.
Self-check: Update all cells from the old row and include the wraparound neighbor.
Novel transfer
Reveal reasoning and rubric
Show that the next average is sufficiently determined by current density in the intended conditions, or add variables such as spatial distribution and boundary flows. A matched average alone does not establish predictive closure.
Self-check: Identify a hidden arrangement, explain its possible effect and state what a reduced model must preserve.
On a later day, try again and record actual evidence in the curriculum. A later unaided explanation and a fresh transfer problem give stronger evidence than immediate familiarity. No reminder is scheduled.
Sources & scope
What supports the lesson?
- David MacKay, Information Theory, Inference, and Learning Algorithms, chapters 2, 4 and 5. Entropy, conditional entropy and the source coding bound.
Original teaching examples. Reference links need a connection; the lesson itself does not. Built 2026-10-11. Learner understanding is not assessed.