Matpic
mathematics, in pictures
Concept

Iteration

Applying a rule to its own output, over and over, and asking what the resulting sequence does.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

xf(x)x ↦ 3.2x(1 − x), started at 0.2the orbit settles into a cycle of 2 points

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.

dynamics · iteration r = 2.6slope in (−1, 1) — attractingr = 3.3slope outside (−1, 1) — repellingat r = 2.6 the slope at the crossing is -0.60 and the staircase walks inat r = 3.3 it is -1.30 and the staircase walks out — the crossing has not moved, its steepness has

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.

dynamics · fixed points step 14two orbits started 0.0001 apart, which is a distance no drawing can showthey are visibly apart by step 14, and by the end share nothing but their interval

A difference too small to draw

Two starting points a ten-thousandth apart, under the same rule, with nothing random anywhere. Within forty steps they have nothing in common — and the rule was not doing anything to them that it does not do to everything.

dynamics · sensitive dependence every c for which the orbit of zero stays bounded, contoured by how long the rest take to escapethe boundary is where the two behaviours meet, and no amount of zoom makes it simple

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.

dynamics · complex numbers the Julia set of c = -0.123 + 0.745ithe same iteration as the Mandelbrot set, with c held still and the starting point varied instead

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.

dynamics · complex numbers root 1root 2root 3each point coloured by which of the three roots Newton's method takes it tothe boundary is not three sectors: every point of it has all three basins arbitrarily close by

Where Newton's method goes instead

An algorithm designed to find roots, run from every starting point at once. Three roots, three basins, and a boundary at which all three are arbitrarily close — so a rule with no randomness in it has starting points whose answer cannot be predicted.

dynamics · newton basins rule 90: the eight neighbourhoods, read as the bits of 90and this is Pascal's triangle modulo two, checked cell by cell against the binomial coefficients

Eight rules and a triangle

A row of cells, each one deciding its next state from the three above it. Eight cases, one bit of output each — a rule that fits in a byte, and 256 of them in total. One of those bytes draws Pascal's triangle.

dynamics · cellular automata rule 110: the eight neighbourhoods, read as the bits of 11060 rows from a random row

The rule that computes

One of the 256 elementary rules can run any program. Not simulate one, not approximate one — a machine that can compute anything computable, built from a lookup table with eight rows and nothing else.

dynamics · cellular automata 0123456721 steps of a rotation by φ − 1 of a turnthe gaps between neighbouring points take 2 distinct values — never more than three, at any number of steps

Three gaps and no more

Turn a circle by the same irrational angle over and over. The points never repeat and never settle, and yet at every single stage the gaps they leave take at most three different lengths — never four, at any number of steps, for any angle.

dynamics · golden ratio 0123456740 steps of a rotation by √2 − 1 of a turnthe gaps between neighbouring points take 3 distinct values — never more than three, at any number of steps

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

dynamics · pigeonhole 9232127 reaches 1 in 111 steps, after climbing to 9232the vertical scale is logarithmic, because the climb is the interesting part and it is steep

The question nobody can answer

Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.

dynamics · collatz

Named alongside it

The objects these essays reach for when they reach for this one.

OrbitComplex numbersFractalLogistic mapSelf similarityBasin of attractionCellular automatonCobwebConnectednessConvergenceDerivativeDeterminism

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