Generator

Rotating by φ − 1 of a turn, 21 times

A generator in the dynamics library, called 65 times across 12 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

circle-map 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

Rotating by φ − 1 of a turn, 21 times. Points on a circle produced by repeatedly turning through the same angle.

The coordinate change that makes an unlocked map a rotation

The images of one interval that never comes back

Two loops of equal area, and the two points where they cross

The same twist without area preservation, and no fixed point anywhere

Orbits of the twist map, unrolled

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.

the orbit 2/3 is found at k = 0.9 ×43
inside the plateau at Ω = 0.4645 there are two orbits ×10
the residue of 3/5 crosses a quarter between 0.9 and 1.05 ×8
the nonlinearity is between 0 and 0.95 ×6
a periodic orbit with rotation number 0/1 exists ×5
the orbit of 0/1 closes ×5
some parameter carries a period-2 orbit with rotation number 1/2 ×3
the rotation number there is exactly 1/2 ×3
inside the plateau at Ω = -0.1528 there are two orbits ×2
the number of parameters sampled is a whole number between 60 and 900 ×2
a whole partial quotient between 2 and 100,000 ×1
above it they grow without bound ×1
above the critical line the interval opens ×1
above the critical line the lift turns back on itself ×1
and at least one is outside it ×1
and both boundary circles still stay where they are ×1
and increasing everywhere else ×1
and it rises rather than falling ×1
and the reason is that this map does not preserve area ×1
and the repelling one's is at least one ×1
and the second really is outside it ×1
and they turn in opposite directions ×1
and would change an exponent-one total a great deal, so that one does not ×1
at it they hold near a quarter ×1
at least two of the panels are inside the plateau ×1
below it they shrink by (1 + √2)² ×1
below it they shrink by the square of the golden ratio ×1
below the critical line every interval is a single number ×1
below the critical line the gap ratio is the square of the ratio ×1
below the critical line they shrink by the golden ratio ×1
below the critical strength the residues shrink to nothing ×1
both boundary circles stay where they are ×1
both maps give the same ratio ×1
both solve from a standing start ×1
doubling the range barely changes this total, so it converges ×1
each crossing is a fixed point of the map ×1
each orbit is followed for between 60 and 600 steps ×1
every conjugacy drawn solves and is increasing ×1
every observed average advance lies in the interval ×1
every rescaled point lies inside the drawing ×1
how many images are drawn either way is a whole number between 6 and 20 ×1
in the sorted coordinates the map is the rotation, to the orbit's own resolution ×1
inside a plateau the rotation number does not depend on where the orbit started ×1
magnified twice more, the staircase looks the same ×1
most sampled parameters carry an orbit ×1
Newton's method closes at every K ×1
Newton's method closes at every K drawn ×1
Newton's method closes for every series drawn ×1
no orbit leaves the annulus ×1
no point of the annulus comes near to staying where it is ×1
on the critical line the gaps shrink by 2.8336 ×1
on the critical line the silver gaps shrink by about 6.799 ×1
once the ones take over, the golden number returns ×1
one fixed point is a centre and the other a saddle ×1
one orbit attracts and the other repels ×1
one to four nonlinearities below one ×1
one to three strengths ×1
outside, the rotation number has moved off the rational ×1
some start near the middle advances at the golden rate ×1
the advance sticks at rational values over whole stretches of starting momentum ×1
the angle is a number or one of the named constants ×1
the angular shift changes sign across the annulus at every angle ×1
the attracting orbit is most strongly attracting in the middle of the plateau ×1
the attracting orbit's multiplier is at most one ×1
the boundaries still turn in opposite directions ×1
the circle wrinkles more as k grows ×1
the closest returns shrink by 1.2886 on both maps ×1
the conjugacy's strip is never wider than the map's ×1
the conserved quantity stays put along every orbit drawn ×1
the critical maps are flat at θ = 0 ×1
the crossings close in on 0.9716 ×1
the drawn images account for less than the whole circle ×1
the drive is between 0 and 1 ×1
the envelopes sandwich the lift ×1
the exponent the lengths fall at is between 1.05 and 4 ×1
the first parameter really is inside the plateau ×1
the gap ratios or the return ratios ×1
the gaps take at most three distinct values, as the three-distance theorem says ×1
the golden divisors never fall below about one over k ×1
the kick is between 0 and 1.5 ×1
the largest nonlinearity is between 0.4 and 1.2 ×1
the largest period is a whole number between 2 and 9 ×1
the length of the orbit is a whole number between 200 and 3000 ×1
the lengths sum to something finite ×1
the locked share of the parameter axis grows with the nonlinearity ×1
the loop and its image cross exactly twice ×1
the loop and its image enclose the same area ×1
the lower end never exceeds the upper ×1
the lower parameter is between 0 and 1 ×1
the map locks onto one step in two over an interval of parameters ×1
the map preserves area everywhere on the annulus ×1
the measured curve is flat over a stretch at more than one rational ×1
the number of drives sampled is a whole number between 40 and 300 ×1
the number of nonlinearity levels is a whole number between 6 and 40 ×1
the number of parameters per level is a whole number between 40 and 320 ×1
the number of starting points is a whole number between 100 and 4000 ×1
the number of steps drawn is a whole number between 10 and 60 ×1
the number of steps is a whole number between 3 and 120 ×1
the option is one this view reads ×1
the orbit leaves no large gap ×1
the orbit repeats exactly when the angle is rational ×1
the outward push is a sensible size ×1
the parameter reaches the intended rotation number ×1
the period is a whole number between 1 and 4 ×1
the perturbation is a sensible size ×1
the perturbation is between a fiftieth and two fifths ×1
the plateau has positive width ×1
the ratio of consecutive lengths rises towards one, which is the smoothness condition ×1
the returns shrink by about 1.29 per Fibonacci step ×1
the rotation number never falls as the parameter rises ×1
the rotation number p/q is between nought and one ×1
the rotation per period, plus one is a whole number between 1 and 5 ×1
the search interval contains the parameter ×1
the tangent map round the orbit preserves area ×1
the two families approach the same silver number ×1
the two orbits are closer together near the edge of the plateau ×1
the upper parameter is between 0 and 1 ×1
the view is one the family draws ×1
the window runs upward and is wide enough to draw ×1
two to six leading twos ×1
up to four settings, each below the critical line ×1
well above it most intervals are open ×1
with no nonlinearity the curve is a straight line and has no plateaus to speak of ×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.

Dynamics

A bounce is a fold of the table

Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.

Dynamics

A rotation in different coordinates

At an unlocked parameter the map is not merely like a rigid rotation; it is one, after a change of coordinates built out of a single orbit. The theorem needs the map to be smooth enough, and the map that shows why is one whose orbit leaves a hole.

Dynamics

A twist that cannot avoid two points

Turn the two edges of a ring in opposite directions without changing any area, and something in between must stay exactly where it is — not one point, but at least two, and the reason is that two loops enclosing the same area have to cross.

Dynamics

A whole interval of speeds

Below the critical line every orbit of the circle map goes round at the same average speed. Above it the map folds back on itself, and the speed depends on where the orbit starts — not a few different values but a whole interval of them, every fraction in it the speed of some periodic orbit, and almost none of them ever seen by an orbit started at random.

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

How smooth the disguise is

An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.

Dynamics

The last circle to break

Kick a spinning rotor once a turn and most of its motions stay on curves that wind round forever, walls no orbit can cross. As the kick grows the walls break one by one, and the last to go is the one whose winding is the golden ratio — at a kick of 0.9716, found by watching a sequence of periodic orbits approximate it and asking whether they are stable.

Dynamics

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

Dynamics

The silver mean at the flat point

At the golden-mean rotation number, the circle map with a flat point has two universal constants, 2.8336 and 1.2886. Repeat the measurement at the silver mean, √2 − 1, and two different numbers appear — 6.7992 and 1.5868 — shared by every map with the same kind of flat point. Below the critical line the silver gaps shrink by (1 + √2)², the square of the silver ratio, exactly as a rigid rotation predicts; on it they do not. Every rotation number with a repeating continued fraction carries its own pair, and a number whose expansion ends in ones inherits the golden pair, whatever came first.

Dynamics

The staircase that is flat almost everywhere

A map of the circle advances by an average amount each step. Plot that average against the parameter driving it and the graph is flat over an interval at every rational, rises only on a set of measure zero, and still climbs from nothing to one.

Dynamics

Three gaps and no more

Turn a circle by the same irrational angle over and over. The points never repeat and never settle, and yet at every single stage the gaps they leave take at most three different lengths — never four, at any number of steps, for any angle.

Dynamics

Two constants at the flat point

Turn up the circle map until its graph goes flat at one point, then hunt for the setting at which an orbit turns by the golden ratio. The settings that lock to 1/2, 2/3, 3/5, 5/8 close in on it, each gap 2.8336 times smaller than the last, and the orbit's returns to its start close in by 1.2886 at every step. Below the flat line the same two numbers are the golden ratio's own. Change the map to a different one with the same kind of flat point and both numbers come back unchanged: they belong to flatness, not to the sine.

The whole library · What the figures prove