The collection

Every essay — page 35

Page 35 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.

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

The standard map as its kick grows. Three square phase portraits of the standard map at increasing strengths, dotted with orbits: curves spanning the square at the smallest, fewer at the critical value, and a scattered sea with islands at the largest.

The last circle to break

Writing 0.3 in base φ. The graph of x ↦ βx mod 1 for the golden ratio φ, the diagonal, and the cobweb staircase of the orbit of 0.3, beside the 8 digits it writes.

Counting in a base that is not a whole number

Where x ↦ βx mod 1 spends its time, the golden ratio φ. A histogram of 2000000 orbit points of the β-map for the golden ratio φ in 50 bins, with Parry's staircase density drawn over it; they agree, and the density steps down at the orbit of 1.

Where a base-β orbit spends its time

The Collatz map on the 2-adic integers, before and after changing to parity coordinates. Two scatter plots of 2048 points: the Collatz map on 2-adic integers in binary-digit coordinates, a scattered cloud, and the same map in parity-vector coordinates, where every point lies on the two lines of the doubling map.

Where the Collatz map is a coin

5 figures $PRIMEGAME: fourteen fractions whose powers of two are the primes. The size in binary digits of the numbers produced by Conway's PRIMEGAME from 2 over 40000 steps, with the pure powers of two marked; their exponents are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.$

Fourteen fractions that list the primes

Where one Hénon orbit spends a million steps. A density map of 1000000 steps of one Hénon orbit counted into 420 × 140 cells: 2542 cells visited, half the time spent in 608 of them.

Where the time goes on an attractor

A quadrilateral the Hénon map carries strictly inside itself. Hénon's trapping quadrilateral with corners (-1.33, 0.42), (1.32, 0.133), (1.245, -0.14), (-1.06, -0.5), the image of its boundary under the Hénon map lying strictly inside it, and an orbit on the attractor.

A region the orbit cannot leave

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.

Every window is the whole diagram again

How many windows of each period, on two maps and as a count of necklaces. period 1: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 2: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 3: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 4: 2 (logistic), 2 (sine), 2 (necklaces), 2 (polynomials); period 5: 3 (logistic), 3 (sine), 3 (necklaces), 3 (polynomials); period 6: 5 (logistic), 5 (sine), 5 (necklaces), 5 (polynomials); period 7: 9 (logistic), 9 (sine), 9 (necklaces), 9 (polynomials); period 8: 16 (logistic), 16 (sine), 16 (necklaces), 16 (polynomials); period 9: 28 (logistic), 28 (sine), 28 (necklaces), 28 (polynomials); period 10: — (logistic), — (sine), 51 (necklaces), 51 (polynomials); period 11: — (logistic), — (sine), 93 (necklaces), 93 (polynomials); period 12: — (logistic), — (sine), 170 (necklaces), 170 (polynomials); period 13: — (logistic), — (sine), 315 (necklaces), — (polynomials); period 14: — (logistic), — (sine), 585 (necklaces), — (polynomials).

Windows counted like necklaces

Traffic with random dawdling: jams from nowhere at density 0.18. Nagel–Schreckenberg traffic, 29 cars on 160 cells, top speed 5, dawdling probability 0.25; up to 11 cars stopped at once.

A jam that comes from nowhere

Rule 30 made reversible: forward 60 steps, then back to the start. Second-order rule 30 on 101 cells: 60 steps forward, then the same rule from the swapped last two rows returns the starting row exactly.

A rule that remembers one row back

The coordinate change that makes the map a rotation. Graphs of the departure from the identity of the conjugacy between the circle map and the golden rotation, at three nonlinearities, each smooth and growing steeper with K.

How smooth the disguise is

Two numbers in one jagged record

Five random Fibonacci sequences growing at one rate. Five jagged lines of log|tₖ| rising with k to a thousand, around a straight line of slope log 1.132, beneath a steeper dashed line of slope log 1.618.

A growth rate no step contains

5 figures $Base-φ digits of two fractions and two irrational numbers. Four rows of 72 base-φ digits each, for 1/3, 2/7, √2 − 1 and 1/π; the two fractions repeat with periods 8 and 16, the two irrational numbers show no period.$

Fractions repeat and roots look random

A ball in a cube and the word of walls it hits, direction (1, √2, √3). A cube drawn in perspective with a billiard path of 30 bounces inside it, each bounce point coloured by the pair of walls it is on, and the word ZYXZYZXYZZXYZYXZYZXYZZYXZYZXYZ below.

The word a line spells in a cube

Rule 30 from one cell, 120 steps, and its centre column. The rule 30 triangle grown from a single cell for 120 rows, with the centre column outlined; it begins 110111001100010110010011.

The column rule 30 will not explain