Mandelbrot set
Named by 8 essays across 2 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.
The folds that measure chaos
Apply the logistic map six times and its graph goes up and down 38 times at r = 3.5 and 64 times at r = 4. How fast that number of folds multiplies with each further step is the map's topological entropy: zero through the whole cascade of period doublings, log of the golden ratio in the window of three, log 2 at the top — and it never decreases as r rises.
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.
Small Mandelbrot sets among the cubics
For most cubics Newton's method finds a root from almost every start. For some it does not: a whole region of starting points falls into a cycle and never finishes. Which cubics are the bad ones is decided by a single starting point — the one critical point that is not a root — and when the cubics are laid out in a plane and coloured by where that point goes, the bad ones form small, exact copies of the Mandelbrot set.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Periodic orbitConformal mapJulia setBasin of attractionBinary expansionComplex numbersConnectednessConvergence rateDoubling mapEscape-timeFractalIteration