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Series

Fixed points — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A map of the interval must fix a point. a continuous map of the interval, drawn with the diagonal. Every continuous map of the interval into itself meets the diagonal somewhere; this one does so at x = 0.6944.

    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.

    part 1 · topology
  2. The field that cannot be combed. A tangent field on the sphere, flowing along the meridians. Every arrow is tangent to the surface, and at the two poles there is no direction for an arrow to take — the field is zero there, and no rearrangement removes both zeros.

    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.

    part 2 · topology
  3. A fixed point that attracts, and one that does not. The same map at two parameters, with the staircase walking towards the crossing in one and away in the other.

    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.

    part 3 · dynamics
  4. A three-coloured triangulation, and the walk that finds a rainbow triangle. A triangle cut into 36 smaller ones, its corners coloured under Sperner's rule. The 9 small triangles carrying all three colours are shaded, and a path enters through a door on one edge and ends inside one of them.

    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.

    part 4 · discrete
  5. x ↦ cos x: two starts, one destination. A map whose graph is nowhere steeper than a fixed factor under one, with staircases from two different starting points converging on the same crossing, and the distance to it falling under a geometric bound.

    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.

    part 5 · analysis
  6. Three sets where a fixed point escapes, and one where it cannot. A ring turned about its centre, an open disc halved toward a point of its rim, the plane shifted sideways, and the closed disc turned and shrunk. Only the last has a point that its map leaves where it is.

    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.

    part 6 · topology
  7. Two loops of equal area, and the two points where they cross. An annulus with the loop of points whose angle is unchanged by the map and the image of that loop, drawn both on the annulus and unrolled into a rectangle. The loops cross at two points, which are the fixed points.

    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.

    part 7 · dynamics
  8. A rule that meets the diagonal, and the same rule with a hole in it. Two panels, each the graph of a rule assigning a set of values to every point of the unit interval, drawn against the diagonal: the first meets the diagonal at a filled-in jump, the second has the jump left open and misses 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.

    part 8 · topology

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