Matpic
mathematics, in pictures
Theme

Small rules, large behaviour

Rules short enough to write on one line, producing behaviour nobody can summarise — and the finding that the size of a rule predicts nothing about the difficulty of the questions it raises.
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 Dynamics

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.

 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 Dynamics

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.

 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 Dynamics

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.

 rule 110: the eight neighbourhoods, read as the bits of 11060 rows from a random row Dynamics

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.

 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 Dynamics

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.

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