Generator

The Mandelbrot set

A generator in the dynamics library, called 44 times across 9 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.

complex-set 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

The Mandelbrot set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region.

A disc that turns for ever

The denominators the series must survive

Reading a radius from the coefficients

The same map at three rotation numbers

Rotation number against the disc it allows

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.

b0 agrees with the value computed by hand ×6
a longer wait only ever decides more ×1
along 1/q the ratio has settled ×1
and after sixteen thousand terms it is still far above the counting estimate ×1
and always straddles 1.5066 ×1
and neither is four tenths ×1
and so is minus one ×1
at the root of the p/q bulb the fixed point's multiplier is exactly e^(2πip/q) ×1
both rays of each pair end, at depth 160, within two hundredths of the bulb's root ×1
each pair's rays end together, beside a centre of exact period n ×1
every angle finds a partner ×1
every bulb up to period forty is within a fifth of sin(πp/q)/q² ×1
every bulb up to period thirteen turns its cycle by its own fraction ×1
every bulb up to period thirty is drawn ×1
every term lowers the bound ×1
every vanishing coefficient has an index divisible by 4 ×1
exactly one cycle of the doubling map turns p places of q ×1
exactly two of its angles are neighbours, one step of 1/(2^q − 1) apart ×1
for every fraction with denominator up to fourteen the landing angles are P01 and P10 ×1
for every pair of neighbouring fractions, the largest bulb between them is their mediant's ×1
most pairs end a thousand times closer to each other than to any other ray ×1
of the 132 non-crossing pairings of the twelve angles, two also clear the chords of periods two and three ×1
on period four, 6/15 is refused 7/15 and 8/15 and joined to 9/15 ×1
one is not ×1
round the fixed point, each step of the cycle advances p places of q ×1
summed over every period the limbs take all the angles ×1
the 235 chords to period eight cross nowhere ×1
the bulb is p/q in lowest terms with q from 3 to 9 ×1
the bulb's outline is found at every multiplier ×1
the centre of the p/q bulb has period q, for every bulb up to period twelve ×1
the coefficients shrink a little faster than one over m ×1
the critical point is in this Julia set exactly when its parameter is in the Mandelbrot set ×1
the escape time separates into bands ×1
the first coefficients to vanish are b4, b8, b12, b16, b24, b32 ×1
the fraction is in lowest terms with q from 2 to 7 ×1
the iteration cap is a whole number between 20 and 400 ×1
the Julia parameter is a complex number ×1
the landing angles differ by exactly 1/(2^q − 1) ×1
the limbs' angles add up, period by period, to φ(q)/(2^q − 1) ×1
the option is one this view reads ×1
the pairs of every period number its centres, and the pairs on one cycle number the satellite bulbs ×1
the pairs that share a cycle end far wider apart than the pairs that do not ×1
the ray of angle 1/6 approaches i geometrically ×1
the ray of angle 1/7 approaches the root like one over the depth ×1
the refusals grow with every period, past twenty for each pair at period twelve ×1
the sample count is a whole number between 60 and 700 ×1
the set is the Mandelbrot set or a Julia set ×1
the span is between 0.0005 and 4 ×1
the undecided cells straddle the area ×1
the undecided share shrinks with every refinement ×1
to period seven, every pair's rays end closer to each other than to any other ray, beside its own centre ×1
towards the golden angle the bulbs outgrow the law; along 1/q they do not ×1
zero is in the Mandelbrot set ×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.

Algebra

A disc that turns for ever

At each fraction on the Mandelbrot set's cardioid a bulb is born. At the irrational numbers between them something rarer can happen: a disc round the fixed point rotates rigidly for ever, with no point falling in or flying out. Whether it exists depends on nothing but the digits of one continued fraction, and on one sum of their logarithms.

Dynamics

An area that never finishes

Newton's method's famous failure is a boundary, and a boundary has no area — a random start misses it with probability one. The real failure is different in kind: a polynomial with small whole-number coefficients whose method has a region of starting points, with area, from which it provably never terminates.

Algebra

Chords that never cross

Every component of the Mandelbrot set's interior has a root, and exactly two external rays land there, with angles that repeat under doubling. Which two can be found without drawing the set at all: list the angles of each period in order, and join each to the next one whose chord crosses no chord already drawn. The rule makes 63 pairs at period 7 and 2,010 at period 12, one for every component; traced into the plane, every pair's two rays end together, beside a centre of the right period and no other.

Dynamics

One c, one picture

The same iteration, with the parameter held still and the starting point varied instead. Every complex number gets its own picture, and moving the parameter a hair can shatter it into dust.

Algebra

The area of the Mandelbrot set, from two sides

The Mandelbrot set has an area, and nobody knows it. Counting grid cells closes in on 1.5066 from both sides, but every count judges a cell by a single point, and points on the boundary never declare themselves. An exact formula exists — π times one minus a sum over the coefficients of the map onto the set's outside — and every partial sum is a guaranteed upper bound; after sixteen thousand coefficients the bound is still 1.786, and at the rate it falls it would need something like 10²⁵ more.

Algebra

The fraction written on every bulb

Round the edge of the Mandelbrot set's cardioid hangs one bulb for every fraction between nought and one. The fraction is the bulb's address, its period, and the turn its cycle makes round a fixed point, all at once, and it sets the bulb's size as well: sin(πp/q)/q², which 245 computed bulbs obey to within fourteen per cent.

Dynamics

The shape in every picture of itself

One line of arithmetic, repeated, with a single complex number as its only input. Sort the numbers by whether the result stays bounded and the boundary between the two answers is the most complicated object anyone draws from a rule this short.

Algebra

Two rays for every fraction

Seen from far away, every point of the Mandelbrot set's outside has an angle, and the curves of constant angle come in to land on the set. At the root of the p/q bulb exactly two of them land, and their angles are written by the staircase under a line of slope p/q. The limbs those pairs fence off share out the whole circle of angles between them, by the identity Σ φ(q)/(2^q − 1) = 1.

Dynamics

Where Newton's method goes instead

An algorithm designed to find roots, run from every starting point at once. Three roots, three basins, and a boundary at which all three are arbitrarily close — so a rule with no randomness in it has starting points whose answer cannot be predicted.

The whole library · What the figures prove