Fixed point
Named by 34 essays across 10 fields — each of them below, with the objects they name alongside it.
Something always stays put
Stir a cup of coffee however violently and let it settle. Some molecule is exactly where it started. Crumple a map and drop it on the region it depicts, and one point lies over the place it names.
Nothing on a sphere can be combed flat
Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.
The staircase that shows the whole orbit
Take a number, feed it to a rule, feed the answer back in. There is a way of drawing that on the rule's own graph which turns the entire future of a starting point into a shape — and the shape is legible.
A point that pulls, and a point that pushes
Every crossing of a curve with the diagonal is a value the rule leaves alone. Whether anything ever arrives there is decided by one number — the slope at the crossing — and the picture makes the reason obvious.
Two lobes and no cycle
Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.
The sentence that says it has no proof
Number every sentence and every proof, and a formal system can talk about itself. Then the diagonal is available one more time, and what it builds is a sentence that is true exactly when it is unprovable.
The rule that forgets where it came from
A walk between a few states, with the next step decided by the current one and nothing else. Run it long enough and the starting point stops mattering — but only when two conditions hold, and both of them have a picture in which they fail.
One line that halves them both
Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.
Three colours force a triangle
Cut a triangle into small ones and colour the corners under one restriction. However the cutting and the colouring are done, some small triangle ends up with all three colours — and the number of them is always odd.
A map that shrinks everything
One extra hypothesis — that every distance is shortened by at least a fixed factor — turns the existence of a fixed point into its uniqueness, an algorithm for finding it, and a bound on the error after any number of steps.
Where the fixed point escapes
The theorem asks for a set that is closed, bounded and free of holes. Drop any one of the three and a map appears that moves every single point — and in each case the point that should have stayed still can be seen leaving.
The shape that averaging leaves alone
Adding independent quantities blurs their distributions together, and rescaling restores the width. Almost every shape is changed by that operation. Exactly one is returned unaltered, and that is why sums of unrelated things keep arriving at it.
A twist that cannot avoid two points
Turn the two edges of a ring in opposite directions without changing any area, and something in between must stay exactly where it is — not one point, but at least two, and the reason is that two loops enclosing the same area have to cross.
How fast the staircase arrives
The slope at a crossing decides whether an orbit reaches it. The same number decides how fast — and when the slope is zero the arithmetic changes kind, from a fixed factor per step to a doubling of the correct digits.
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.
Necessity that means provable
Read the box as "the theory proves" and one modal logic stops being a proposal about what necessity might mean. It becomes a complete description of what a formal system can prove about its own proofs — and its frames run forward, compose, and stop.
Every site in the middle of its own cell
Move each point to the centre of mass of its own Voronoi cell, then redraw the diagram, then do it again. The rule is two lines long, it never mentions hexagons, and what it settles into is a honeycomb.
The sphere that complex numbers live on
Add one point to the complex plane and it becomes a sphere. The rotations of that sphere are exactly the maps written as one linear expression divided by another, so a fact about turning a ball is a fact about dividing polynomials.
The flow that is really a map
A trajectory wandering through three dimensions is hard to reason about. Record only the successive maxima of one coordinate and the wandering collapses onto a curve — a map of an interval to itself, with a corner in the middle, which is a thing the theory can handle.
The moment a giant appears
Raise the chance of an edge slowly and a random graph does nothing for a long time, then in a narrow window acquires a component holding a definite fraction of everything. The fraction is the root of an equation, and the equation says why the transition is where it is.
How many get their own hat
The chance that nobody gets their own hat settles on 1/e. The chance that exactly one person does settles on 1/e too, exactly two on 1/(2e), exactly three on 1/(6e) — the Poisson distribution with mean 1. The reason is a set of averages that come out exactly 1 at every size, and the counts reach the limit so fast that eight hats are within six ten-thousandths of it.
A map that offers a choice
Brouwer's theorem needs a function, and the object it was most wanted for is not one — a best reply is a whole set whenever a chooser is indifferent. Allow a point to be sent to a set and the fixed point survives, provided the sets are convex, and the convexity is the entire hypothesis.
The window that opens with a stutter
The period-three window does not fade in. At r = 1 + √8 a cycle of three appears out of nothing, and just before it does, the chaotic orbit keeps imitating the cycle that is not there yet — for twenty steps, then fifty, then hundreds, in quiet stretches whose length grows as one over the square root of the distance to the window.
The word that cannot describe itself
Some adjectives describe themselves — 'short' is short — and some do not — 'lengthy' is not lengthy. Call the second kind heterological, and ask whether 'heterological' is heterological. It is exactly when it is not. The diagonal that beat every list of numbers, every set of sets and every provability predicate has been turned on words that describe words, and it proves that no language can contain a word for its own notion of describing, naming or truth.
A rent nobody envies
Three housemates, three rooms that are not alike, one rent. Every way of splitting the rent is a point of a triangle; ask, at each point of a fine grid, which room one housemate would take at those prices, taking turns so that each small triangle has one corner for each of them. Sperner's lemma then promises a small triangle where all three would choose different rooms — and as the grid is refined, the envy at that triangle shrinks to nothing.
The program that prints itself
The diagonal argument has always been used to destroy — to show that a list misses something, that a sentence cannot be proved. Run the same move the other way and it builds. Kleene's recursion theorem says every program can be given its own text to work with, and the proof is a program that prints itself, thirty-two characters long, which can be run and checked.
What it costs to find the point that stays put
Brouwer's theorem promises a point a map leaves where it is, and Sperner's corridor walks to it. On a gentle map the walk is short; twist the map about its fixed point and the corridor follows every turn, so its cost grows like the square of the grid. Halving the square by winding number finds the same point in a number of readings proportional to the grid's side whatever the map does — and no method that only reads the map can do fundamentally better.
Square it and keep the middle
John von Neumann's first generator of random numbers squared a number and kept its middle digits. Followed from every four-digit seed, it never lasts more than 111 steps before a value repeats — a third of what the birthday problem allows a truly random rule — and one seed in five ends at zero for ever. A counter added before each squaring cures the collapse.
A cut that gets sharper as the map gets lazier
Iterating a map that shrinks distances by 0.9999 takes 141,254 steps to reach its fixed point to a millionth. Cutting takes six: read the map at one point, and the fixed point must lie on the far side of the line halfway between the point and its image. The cut gets better exactly where iteration gets worse, because a map that barely shrinks puts its fixed point almost on every such line.
A board that cannot be drawn
Colour a rhombus of hexagons in two colours. One colour must join its two sides, and only one: in all 65,536 colourings of the four-by-four board, no exception. The proof is a walk along the colour boundary, the same walk that proves Sperner's lemma, and David Gale showed that the no-draw theorem of this board game is equivalent to Brouwer's fixed-point theorem. Nobody knows the winning strategy for a full-sized board, though a short argument proves one exists.
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.
Root two, raised to itself for ever
Stack √2 to the power √2 to the power √2 without end and the tower settles at exactly 2. Solve the equation for a tower equal to 4 and √2 appears again, and that answer is false. The bases that give a limit run from e^(−e) to e^(1/e), the slope at the crossing decides everything, and a sum over labelled trees turns out to be the same number.
A number that counts what it cannot see
Whether a map has a fixed point seems to need the map. Lefschetz's number needs only what the map does to holes, and it stays the same however the map is bent. On a torus it is one determinant: |det(I − A)| fixed points, checked exactly on all 2,401 small matrices, 1, 5, 16, 45, 121 periodic points for Arnold's cat, and a formula that also counts solutions of equations over finite fields.
Named alongside it
The objects these essays reach for when they reach for this one.
BrouwerIterationExistence proofConvergence rateNonconstructiveOrbitWinding numberContinuityConvergenceInvariantPeriodic orbitSelf-reference