Elementary cellular automaton, rule 90
automaton is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
Rule 184 at density 0.30
The five rules that conserve
Flow against density on a ring road
Jams dying out, and jams that cannot
Rule 90, as a lookup table
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- after one lap the flux at density 0.025 is min(ρ, 1 − ρ) ×40
- rule 90 row 0 matches Pascal's triangle mod 2 ×40
- every configuration of a 5-cell ring ends as its majority ×6
- a density strictly between 0 and 1 ×1
- above half density at least 2k − n cars are always stopped ×1
- an odd ring of 7 to 13 cells ×1
- an odd ring, so there is always a majority ×1
- and freezes with both values still present ×1
- and the drawn ring ends as its majority ×1
- below half density every jam is gone after at most one lap ×1
- below one half the stopped share reaches zero ×1
- every cell is on or off and nothing else ×1
- exactly five of the 256 rules conserve the number of 1s on every small ring ×1
- far from one half it is always right ×1
- near one half it is often wrong ×1
- no elementary rule classifies every ring ×1
- rule 184 never creates or destroys a car ×1
- stopped cars never fall below (2k − n)/n ×1
- the drawn table reads back as the rule number ×1
- the local vote freezes within a few steps ×1
- the number of rows is a whole number between 4 and 160 ×1
- the ring's length is a whole number between 20 and 240 ×1
- the rule number is a whole number between 0 and 255 ×1
- the seed is single or random ×1
- the start is random or one block of cars ×1
- the steps drawn is a whole number between 10 and 160 ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A road where nobody overtakes
Rule 184 moves every 1 one cell to the right whenever the cell ahead is empty. It is one of only five elementary rules that never change the number of 1s, and that single property turns it into a model of traffic with an exact transition: below half density every jam dissolves, above it jams can never all clear and drift backwards against the flow.
DynamicsEight rules and a triangle
A row of cells, each one deciding its next state from the three above it. Eight cases, one bit of output each — a rule that fits in a byte, and 256 of them in total. One of those bytes draws Pascal's triangle.
DynamicsNo local rule can count the votes
A ring of cells, each holding 0 or 1, has to agree on whichever value is in the majority — every cell seeing only its neighbours. The best-known rule gets it right most of the time and wrong near a tie; no rule of any radius gets it right always. Yet two rules run one after the other do, on every ring, and the first of them is the traffic rule.
DynamicsThe orbit written as a word
Cut the interval in two and record which half each step of an orbit lands in. The orbit becomes an infinite string of two letters, the map becomes the act of deleting the first letter, and questions about trajectories turn into questions about words.
DynamicsThe rule that computes
One of the 256 elementary rules can run any program. Not simulate one, not approximate one — a machine that can compute anything computable, built from a lookup table with eight rows and nothing else.