Generator

Rotating by φ − 1 of a turn, 21 times

A generator in the dynamics library, called 24 times across 5 essays. Below: what it draws at its defaults 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.

At its defaults

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

`show: "twist"`

Two loops of equal area, and the two points where they cross. An annulus with the loop of points whose angle is unchanged by the map and the image of that loop, drawn both on the annulus and unrolled into a rectangle. The loops cross at two points, which are the fixed points.

`show: "no-area"`

The same twist without area preservation, and no fixed point anywhere. An annulus whose boundary circles turn in opposite directions while every interior point is also pushed outward, with short segments showing where each sample point goes. Nothing stays put.

`show: "orbits"`

Orbits of the twist map, unrolled. Several orbits of an area-preserving twist map of the annulus, drawn with angle across and radius up: nested curves, a chain of islands around the centre, and the crossing streams of the saddle.

`show: "locked"`

One orbit inside a plateau and one just outside it. Two orbits of a circle map drawn as points round a circle, one settling onto a repeating pair of positions and the other filling the circle.

`show: "staircase"`

The rotation number against the parameter at K = 1. The measured rotation number of a circle map plotted against its parameter, forming a staircase that is flat over an interval at each simple rational.

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 nonlinearity is between 0.4 and 1.2 ×2
and both boundary circles still stay where they are ×1
and the reason is that this map does not preserve area ×1
and the second really is outside it ×1
and they turn in opposite directions ×1
both boundary circles stay where they are ×1
each crossing is a fixed point of the map ×1
each orbit is followed for between 60 and 600 steps ×1
inside a plateau the rotation number does not depend on where the orbit started ×1
no orbit leaves the annulus ×1
no point of the annulus comes near to staying where it is ×1
one fixed point is a centre and the other a saddle ×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 boundaries still turn in opposite directions ×1
the conserved quantity stays put along every orbit drawn ×1
the first parameter really is inside the plateau ×1
the gaps take at most three distinct values, as the three-distance theorem says ×1
the largest nonlinearity is between 0.4 and 1.2 ×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 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 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 parameters sampled is a whole number between 60 and 900 ×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 orbit repeats exactly when the angle is rational ×1
the outward push is a sensible size ×1
the perturbation is a sensible size ×1
the perturbation is between a fiftieth and two fifths ×1
the rotation number never falls as the parameter rises ×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
with no nonlinearity the curve is a straight line and has no plateaus to speak of ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

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 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

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 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.

The whole library · What the figures prove