The Mandelbrot set
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
A Julia set
The Mandelbrot set
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.
- and neither is four tenths ×1
- and so is minus one ×1
- one is not ×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 iteration cap is a whole number between 20 and 400 ×1
- the Julia parameter is a complex number ×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
- 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.
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.
DynamicsOne 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.
DynamicsThe 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.
DynamicsWhere 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.