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mathematics, in pictures
Ladder

Iteration — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. the logistic map at 3.2, iterated from 0.2. A map drawn as a curve with the diagonal across it, and the staircase that iterating it produces.

    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.

    rung 1 · dynamics
  2. How fast an iteration arrives. The distance from the fixed point plotted against the step number on a logarithmic vertical axis, for two ordinary iterations and for Newton's method, whose curve bends downward.

    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.

    rung 2 · dynamics
  3. The tent map and the logistic map, joined by a change of coordinate. Two cobweb diagrams side by side — the tent map at slope two and the logistic map at four — with the orbit of one carried to the orbit of the other by a curve drawn between them.

    The same map in different coordinates

    The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

    rung 3 · dynamics
  4. Where a long orbit of the logistic map spends its time. A histogram of 60000 iterates in 32 bins, with the density the map preserves drawn over it as the exact share each bin should hold.

    The histogram an orbit leaves

    When no single step of an orbit is worth reporting, what is left is where it spends its time. That distribution is not uniform, it does not depend on where the orbit started, and it can be computed in closed form.

    rung 4 · dynamics
  5. The orbit a computer draws, and the orbit. Two orbits of the tent map from the same starting fraction plotted against the step number — one computed exactly in whole-number arithmetic and periodic, one computed in double precision and reaching zero.

    The orbit a computer draws

    A chaotic orbit computed in floating point is not the orbit of the point it started from. Sometimes it is the true orbit of a nearby point, which is enough; sometimes the arithmetic simply runs out, and the picture is of the rounding.

    rung 5 · dynamics

All ladders