Fractal
Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
Curvatures that stay whole
Four circles touching one another satisfy an equation in their curvatures. Read it as a quadratic and the second solution is the first subtracted from something — so a packing that starts with whole numbers stays whole forever.
A cubic method that is Newton's in disguise
Halley's method, from 1694, uses the second derivative as well as the first and cubes the error at every step where Newton's squares it. It is also, exactly, Newton's method applied to a different function — p divided by the square root of p′ — and that single fact explains why its basins are calmer, why it walks out of the trap that holds Newton for ever, and why its boundaries are still fractal.
The ball that stays outside the table
Turn billiards inside out. A point outside a convex table looks at the corner on its right, jumps straight through it, and lands as far beyond as it started before. Round a square every orbit closes; round a circle every orbit keeps to its own circle; round a regular pentagon the orbits form islands with a fractal between them. Whether some table lets a point wander off to infinity was Moser's question, and the answer — yes, for a kite — took until 2007.
Where the time goes on an attractor
A chaotic orbit's position is unpredictable within a few dozen steps. How it divides its time is not — start anywhere, follow long enough, and the fraction of time spent in each region comes out the same. That distribution lives on a set of no area, and among the infinitely many ways an orbit could spend its time, typical starts pick exactly one.
A dust that almost every line misses
Keep the four corner squares of a square, then the four corners of each of those, and so on. What is left has a length, in the sense that measures length, and yet almost every straight line misses it entirely. Stage by stage the chance that a random line hits the dust falls — to 48% by the eighth stage — and how fast it falls to nothing is one of the few questions about a simple picture that is still open.
Named alongside it
The objects these essays reach for when they reach for this one.
IterationBasin of attractionComplex numbersPeriodic orbitBoundednessConnectednessEscape-timeMandelbrot setNewtons methodOrbitRoot-findingSelf-similarity