Conformal map
Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.
A sphere is a plane plus one point
Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.
The map that trades circles for lines
Send every point to the one on the same ray whose distance multiplies with it to a fixed number, and circles become lines, lines become circles, angles survive untouched, and a ring of tangent circles falls out of a ring of equal ones.
Eight circles touching three
Draw three circles. How many circles touch all three? The answer is eight, the count is a fact about signs rather than about geometry, and the classical way to find them is to move the problem somewhere it becomes easy.
The number four points agree on
One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.
Every flat graph is a pile of circles
A graph that can be drawn without crossings can be drawn in one particular way: as circles, one per vertex, touching exactly when their vertices are joined. The picture is not a choice — it is determined, up to the group two inversions generate.
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.
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.
Where a random walk leaves the snowflake
Start a random walk at the centre of the Koch snowflake and record where it first touches the edge. The edge has dimension 1.26, but the places the walks arrive form a set of dimension one: 60,000 walks spread their entropy by ln 3 a level, not ln 4, and the busiest fifth of the edge takes nine tenths of them. Makarov proved in 1985 that it must be so for every simply connected region in the plane.
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.
InversionMandelbrot setBinary expansionCircleCircle-preservingConstructionConvergence rateDoubling mapFixed pointMobius transformationPeriodic orbitTangency