Small rules, large behaviour — page 2
Applied
A mixture that is a population
A mixed equilibrium between two choosers is a knife-edge nobody has a reason to stand on. Read the same mixture as a population whose shares grow with how well they do, and it becomes a point every population is carried to — or one every population circles for ever without arriving.
Dynamics
A matrix that counts the returns
Draw a map straight through the cycle 0 → 1/3 → 1 and its two pieces carry each other in a fixed pattern: the left piece only across the right, the right across both. The orbits' words are then walks on a two-node graph, and the number of points that come back after n steps is the trace of that graph's matrix to the nth power — 1, 3, 4, 7, 11, 18 — each one checked by solving for the points exactly.
Dynamics
The folds that measure chaos
Apply the logistic map six times and its graph goes up and down 38 times at r = 3.5 and 64 times at r = 4. How fast that number of folds multiplies with each further step is the map's topological entropy: zero through the whole cascade of period doublings, log of the golden ratio in the window of three, log 2 at the top — and it never decreases as r rises.
Algebra
How fast the ball fills
Count the elements within r steps of doing nothing. The count grows like a polynomial in some groups and like a power of three in others, the distinction survives every change of generating set, and which polynomial degrees are possible is a theorem nobody expected.
Dynamics
How a lock comes apart
Inside a plateau the map has two periodic orbits, one attracting and one pushing away. Track them to the plateau's edge and they run into each other and vanish together — so the edge of a tongue is a collision, and the width of the plateau is how far the two can be pulled apart.
Dynamics
The dark lines are one point's orbit
Past the end of the period-doubling cascade the bifurcation diagram turns into grey bands crossed by darker curves. Every one of those curves is the orbit of a single point — the top of the hump — and the places where they meet are exactly where the bands merge, in a second cascade that runs backwards at the same rate.
Dynamics
The window that opens with a stutter
The period-three window does not fade in. At r = 1 + √8 a cycle of three appears out of nothing, and just before it does, the chaotic orbit keeps imitating the cycle that is not there yet — for twenty steps, then fifty, then hundreds, in quiet stretches whose length grows as one over the square root of the distance to the window.
Discrete
The few points that cut a flat graph
Any graph that can be drawn without crossings, however large, falls into pieces of at most two thirds once a few points are removed — about the square root of its size, never more than 2.83 times it. A grid shows the square root cannot be beaten, a ring of breadth-first neighbours comes close, and a cycle through a shallow tree finishes the job.
Dynamics
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.
Dynamics
No 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.
Dynamics
A solvable chaos of every degree
The logistic map at four is chaotic and, through a change of coordinates, completely solvable: its orbits are cosines of doubling angles. The trick is not a one-off. For every whole number n there is a polynomial of degree n that multiplies angles by n instead of 2, and every one of them is exactly as solvable, has exactly nᵏ points of period k, and preserves the same distribution — and any two of them commute, which almost no two polynomials do.
Topology
A twist that carries one loop to another
Cut a torus along a loop, turn one side of the cut once round, and glue it back. Nothing is torn, so every loop that did not cross itself still does not — but a loop of class (0, 1) is now a loop of class (1, 1). Two such twists reach every loop that never crosses itself, by Euclid's algorithm, and the symmetries they generate are exactly the whole-number matrices of determinant one.
Discrete
Multiplying every number on the dial at once
Join every residue on a dial to twice itself and the chords draw a heart-shaped curve with one cusp; join each to three times itself and the curve has two. The picture is the whole multiplication map at once, and it holds three facts: the map splits the dial into cycles whose lengths are orders, those cycles on a dial of 2ⁿ − 1 are the binary necklaces of length n, and the curve is the caustic light draws inside a cup.
Dynamics
The ball that stays outside the table
Turn billiards inside out. A point outside a convex table looks at the corner on its right, jumps straight through it, and lands as far beyond as it started before. Round a square every orbit closes; round a circle every orbit keeps to its own circle; round a regular pentagon the orbits form islands with a fractal between them. Whether some table lets a point wander off to infinity was Moser's question, and the answer — yes, for a kite — took until 2007.
Dynamics
The word a straight line spells
A ball rolling across a square table, forever, hits walls in some order: V for a side wall, H for the end wall. Unfold the table and the ball becomes a straight line across a grid, and the order of walls becomes a word — VHVHVVHVHVVHV… for the golden slope. That word has exactly n + 1 different blocks of every length n, the fewest any word that never repeats can have; every stretch of it holds its fair share of H's to within one; and its blocks occur with at most three different frequencies.
Geometry
Tiles that never repeat
Two rhombs with angles taken from the pentagon, cut each into halves, and cut each half into smaller copies of the two halves by a fixed rule. Repeat, and the pieces fill the plane with no gaps and no overlaps, in a pattern with five-fold stars everywhere and no period anywhere. The reason it cannot repeat is a single number: thick tiles outnumber thin ones by φ, and a repeating pattern would make that ratio a fraction.
Dynamics
Where the Collatz map is a coin
Extend the Collatz map from the whole numbers to the 2-adic integers — binary strings that run on for ever to the left — and it stops being mysterious. It becomes, after a change of coordinates, the simplest chaotic system there is: shifting a string of coin tosses one place. Everything about it is then known, and none of it says anything about the whole numbers, which is the most instructive failure in the whole story.
Dynamics
Fourteen fractions that list the primes
Change the Collatz rule so that the multiplier depends on the remainder modulo some other number than two, and the resulting maps can compute anything a computer can. John Conway proved it in 1972, which means no method can decide, for every such map, whether its orbits come down — and fourteen fractions, applied in order, turn out to be enough to print every prime.
![A strand of the logistic map's diagram between r = 3.8284 and 3.8572, upside down. The bifurcation diagram of the logistic map cropped to r in [3.8284, 3.8572] and x in [0.44, 0.565], with the value axis inverted.](/f/a-strand-of-the-logistic-map-s-diagram-between-r-3-8284-36b84e6c.svg)
Dynamics
Every window is the whole diagram again
Zoom into one strand of the period-three window of the logistic map and the whole bifurcation diagram is there again — a single value, a fork, a cascade of doublings, chaos, bands and windows — upside down and ninety-seven times narrower. The doublings inside it shrink by Feigenbaum's 4.669, exactly as the originals do, because near the top of its hump the map's third iterate is itself a one-humped map, turned over.
Dynamics
A jam that comes from nowhere
Give cars on a ring road a top speed of five cells a step, let each slow to the gap ahead, and add one more rule: now and then, at random, a driver eases off by one. That is the whole of the Nagel–Schreckenberg model, and it produces what the exactly solvable rule 184 could not — jams that form in free traffic with no obstacle, drift backwards against the flow, and cost the road more than a third of its capacity. Set the top speed to one and remove the chance, and it is rule 184 again, cell for cell.
Dynamics
A rule that remembers one row back
Only six of the 256 elementary cellular automata can be run backwards, and all six are trivial: shifts, copies and complements. Every interesting rule forgets. But make the new row depend on the row before the current one as well — combine the rule's output with it cell by cell — and every rule, rule 30 included, becomes exactly reversible: run forwards, swap the last two rows, run the same rule again, and the starting row comes back cell for cell. Nothing is lost, and yet the patterns still look as disordered as ever.
Probability
An urn forgets its start only below one half
Let each draw from an urn add balls of both colours in fixed amounts, and the long run depends on a single ratio of two eigenvalues. Below one half the urn behaves like a coin, its fluctuations spread like the square root of the draws and settle into a bell. Above one half the first few draws decide most of the outcome, the spread grows faster, and the shape that results is not a bell and depends on how the urn began.
Dynamics
A flow that doubles like a parabola
Otto Rössler's three equations have one nonlinear term and a single knob. Turn it and the orbit closes after one loop, then two, then four, then never — the logistic map's cascade, in a flow in three dimensions. Record each peak of the orbit against the one before and the reason appears: the points lie on a single rounded hump. The gaps between doublings shrink by ratios 3.49, 4.54, 4.59, climbing towards Feigenbaum's 4.669, and the chaos contains a period-3 window that repeats the whole cascade.
Logic
Dominoes that no algorithm can match
Emil Post's puzzle has no machines and no programs in it: a handful of dominoes, each with a string on top and a string underneath, and the question whether some row of them, repeats allowed, spells the same word along both edges. No algorithm can answer it for every set of dominoes, because a set can be built whose only matches are a machine's run written out twice, one step out of phase — and three dominoes with strings of three characters already need thirty-six in a row before the edges agree.