The collection

Every essay — page 34

Page 34 of 50, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Dynamics

One rule, applied over and over, and what the sequence does in the end.

Infinite below 0.6309, nought above it. The total of the s-th powers of the diameters in the natural cover of the middle-thirds Cantor set, plotted against s for 4 depths. Every curve passes through one at s = 0.6309 and they separate either side of it.

Infinite on one side and nought on the other

A carpet whose two dimensions differ by 0.076. A self-affine carpet built by keeping 5 cells of a 4 by 2 grid and repeating 4 times. Its box dimension is 1.6610 and its Hausdorff dimension 1.5850.

A carpet with two dimensions

A dimension of 1.2576, from two stretching rates. The running averages of the Hénon map's two Lyapunov exponents, settling at 0.4177 and -1.6217. Kaplan and Yorke's formula turns them into a dimension of 1.2576 without counting a single box.

A dimension from the stretching rates

A dimension for every rate of crowding

A graph of dimension 1.585, built by raising midpoints. The Takagi–Landsberg graph with w = 0.75 on the unit interval, drawn with 16 columns shaded by how far the graph rises and falls across each. Counting boxes at seven widths gives a slope of 1.582.

The room a jagged graph takes up

A map drawn through the cycle 0 → 1/3 → 1, and the graph its pieces make. The graph of a map made of two straight pieces through a cycle of three points, with the cycle drawn as a staircase, beside a two-node graph showing which piece may follow which and the matrix of that graph.

A matrix that counts the returns

The logistic map applied 6 times at 3.5, 3.83, 4, and its folds. Side-by-side graphs of the logistic map composed with itself, one per parameter, each labelled with how many monotone pieces it has.

The folds that measure chaos

Two periodic orbits meeting at the edge of the 1/2 plateau. The circle at 4 parameter values, with the period-2 points marked — two orbits inside the plateau, drawing closer together towards its edge, and none outside it.

How a lock comes apart

The coordinate change that makes an unlocked map a rotation. The conjugating map built from one orbit, plotted as a staircase from the circle to itself, with the rigid rotation it turns the circle map into.

A rotation in different coordinates

An area of starting points from which z³ − 2z + 2 is never solved. The complex plane coloured by which root of z³ − 2z + 2 Newton's method reaches from each starting point, with the points that reach no root left uncoloured.

An area that never finishes

67 starting points that find all 5 roots. The roots of z⁵ − 1 with a ring of starting points around them, each start marked by which root the method reaches from it, and every root reached by at least one.

Covering rather than avoiding

The turning point's orbit over the bifurcation diagram, 3.55 to 4. The bifurcation diagram of the logistic map from 3.55 to 4 with the curves f(1/2), f²(1/2), … up to the 6th image drawn over it. The first two bound the attractor and the rest trace the dark lines inside it.

The dark lines are one point's orbit

Intermittency at r = 1 + √8 − 0.0003. A time series of 600 steps of the logistic map just below the period-three window. Long stretches that look like a cycle of three, shaded, alternate with irregular bursts; there are 6 such stretches here.

The window that opens with a stutter

Rule 184 at density 0.30. A space-time diagram of rule 184 on a ring of 120 cells, 80 steps down the page, starting from a random row with 36 cars. The diagonal stripes are free-moving cars; the jams dissolve.

A road where nobody overtakes

A local rule taking a vote. A space-time diagram of the GKL rule on 149 cells from a random row with 69 ones. Black and white regions grow and meet along slanting boundaries, and after 69 steps the whole ring is 0.

No local rule can count the votes

6 figures $How long four orbits of the doubling map spend in each quarter. Grouped bars for four starting points of the doubling map, each group showing the fraction of time the orbit spends in each quarter of the unit interval, against a dashed line at one quarter.$

Almost every orbit is fair

The Chebyshev maps T₂, T₃, T₄ and T₅. Four small square plots of the Chebyshev polynomials of degrees two to five on the interval from minus one to one, each with the diagonal drawn; the graph of degree n sweeps between the bottom and top of the square n times.

A solvable chaos of every degree

A periodic point and a wandering one, both near 0.3, parted by step 4. The distance between the orbit of a periodic point and the orbit of a point from a dense orbit, both starting in the same small interval, plotted against the step until the wandering orbit nears the point farthest from the periodic one.

Sensitivity comes free

What the tent of slope 3 keeps: 32 pieces after 5 steps. Rows showing the parts of the unit interval that remain inside it for 0 to 5 steps of the open tent map of slope 3, halving into a Cantor set.

Chaos on a set nobody lands on

How many steps Newton's method needs on (z − 1)²(z + 1). A square of the complex plane shaded in bands by the number of iterations a root-finding method needs from each starting point, darker meaning slower, with the roots marked.

A double root halves the error instead of squaring it

Newton's and Halley's basins for z³ − 1. Two squares of the complex plane side by side, each coloured by which root a starting point converges to, the left under Newton's method and the right under Halley's, with non-converging starts marked.

A cubic method that is Newton's in disguise

Every orbit round a square closes. 5 outer-billiard orbits about a square, each drawn as its closed ring of points, with periods 4, 8, 12, 20, 24 growing outwards.

The ball that stays outside the table

A straight line across a grid, and the word of walls it crosses. A line of slope 0.6180 crossing a unit grid, with each crossing marked V or H, beside the same path folded into a square as a billiard trajectory; the word begins VHVHVVHVHVVHVVHVHVVH.

The word a straight line spells

Two staircases and the interval between them, at K = 1.5. A plot against the drive of the upper and lower ends of the circle map's rotation interval above the critical line, two stepped curves with the band between them shaded.

A whole interval of speeds