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.
The Mandelbrot set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region. 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.

 A Julia set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region. 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.

 Elementary cellular automaton, rule 90. A row of cells evolving downward, each cell decided by the three above it. 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.

 Elementary cellular automaton, rule 110. A row of cells evolving downward, each cell decided by the three above it. 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.

 The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale. 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.

 Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross. Computation

What two points can build

A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.

 The non-zero elements of GF(16) as the powers of one of them. A ring of the field's non-zero elements in the order the powers of a primitive element produce them, beside a table of exponents. Computation

Every element is a power of one of them

Pick the right element of a finite field and its powers run through every other non-zero element exactly once before returning to one. Multiplication becomes addition of exponents, and a table of q − 1 entries replaces the whole multiplication table.

 One matching that is not stable, and all 24 counted by blocking pairs. An unstable matching with its blocking pair ringed and both members' rankings marked, above an exhaustive census of every matching of the instance by how many blocking pairs it has. Applied

Nobody has a reason to run away

A matching is stable when no two people on opposite sides would both rather have each other than what they have — a condition that names nothing to build and everything to rule out. The surprise is that something always satisfies it, however perverse the rankings are made.

 The 4 stable matchings of the instance, ordered by side one's preference. A Hasse diagram of the stable matchings with the best for side one at the top, beside a table giving the join and the meet of every pair. Applied

The side that proposes wins

An instance usually has several stable matchings, and the set of them is not a heap — it is a lattice, closed under taking the better partner and under taking the worse. The two ends of that lattice are exactly what deferred acceptance returns from the two sides, so whoever proposes decides which end the instance lands on.

 A billiard path of slope 0.618, folded and unfolded. A ball bouncing inside a square table, and the same trajectory drawn as one straight line through reflected copies of the table, so that the bounces disappear. Dynamics

A bounce is a fold of the table

Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.

 the Koch curve, after 5 steps. the Koch curve drawn from its own rule: replace the middle third of every segment with two sides of a triangle. Dynamics

A dimension that is not a whole number

Cover a set with boxes of side ε and count how many are needed. For a line the count grows like 1/ε, for a region like 1/ε². For the Koch curve it grows like 1/ε to the power 1.26, and that exponent is as good a definition of dimension as the other two.

 The orbit of 13/32 under doubling, and its word. A cobweb of the doubling map with one orbit drawn, the interval split in half beneath it, and the letter each step contributes written out in order. Dynamics

The orbit written as a word

Cut the interval in two and record which half each step of an orbit lands in. The orbit becomes an infinite string of two letters, the map becomes the act of deleting the first letter, and questions about trajectories turn into questions about words.

 the right triangle at an eighth of a turn: 16 directions, and a surface of genus 2. A polygonal billiard table with a long trajectory drawn on it, the finite set of directions that trajectory takes, and the arithmetic of the surface it unfolds into. Dynamics

A table folded into a surface

Unfolding a square billiard gives a straight line on a torus. Unfolding any table whose angles are whole fractions of half a turn gives a straight line on some surface — and which surface it is decides how hard the dynamics will be.

 A room a trajectory cannot get out of, and one it can. A mushroom-shaped billiard table with two long trajectories: one confined to the cap by a conserved quantity, and one that enters the stem. Dynamics

A room that cannot be lit

Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.

 One disc, and two paths that stop being near each other. Two nearly identical billiard paths drawn on an empty square and on a square with a circular obstacle, with the separation between them plotted against distance travelled. Dynamics

The obstacle that makes a table chaotic

Put one round post in the middle of a square table and every trace of order goes. Two paths that start a hundred-thousandth of a degree apart end up on opposite sides of the table, and the reason is that a wall curving outwards multiplies a gap where a flat one only adds to it.

 A 4-bit register that visits all 15 nonzero states. The first 15 states of a 4-bit linear feedback shift register with taps at 4 and 1, with the bit that leaves the register at each step; the output shows every nonzero window of 4 bits exactly once. Computation

A memory of four bits

A register holding four bits, shifting them along and adding two of them back, runs through all fifteen nonzero states before it repeats. Which two are added back is a question about a polynomial, and getting it wrong costs fourteen of the fifteen.

 A scatter walking towards its own centres. 4 panels of the same 24 sites: the initial clumpy scatter and the Voronoi diagram after 1, 3, 12 rounds of Lloyd's iteration, with the cost falling to 52% of the scatter's as the cells even out. Geometry

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 14 triangulations, joined by single flips. The flip graph of a 6-gon: 14 triangulations drawn as small polygons and joined by 21 edges, one for each pair differing in a single diagonal. Discrete

The solid whose corners are triangulations

Take the triangulations of a hexagon as points and join two of them when a single diagonal can be swapped for another. The result is not merely a graph — it is the edge skeleton of a genuine convex polyhedron, with fourteen corners, three square faces and six pentagonal ones.

 A closer start buys time and nothing else. The logarithm of the separation between two Lorenz trajectories plotted against time, for three different initial separations. The three curves are straight and parallel over most of their length, with the same fitted slope. Dynamics

A closer start buys only time

Two trajectories from almost the same place separate exponentially, and the rate does not depend on how close they began. Halving the initial error buys one fixed interval of extra agreement, and no amount of precision buys more than a fixed number of those.

 Stretch, fold, and what is left. 5 stages of the horseshoe map's surviving set: one square, then two strips, then four, up to 16, each narrower than the last by a factor of 3. Dynamics

Stretch, fold, and what is left

A system that pushes every pair of nearby points apart and keeps them all inside a bounded region has only one option, and it is the one a baker uses. Stretching and folding is the mechanism, and what survives infinitely many folds is a Cantor set.

 The moment a giant piece appears. The largest component's share of 900 points plotted against the average degree, with the measured values as dots and the predicted curve behind them. The curve is flat at zero below an average degree of one and rises steeply above it. Probability

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.

 At the threshold the largest part is neither of the two obvious sizes. A table with one row per graph size, giving the largest component at the critical edge probability and that value divided by three candidate scalings. Probability

The window where the giant is born

Below the threshold the largest piece is a few dozen points, above it a definite fraction of everything. At the threshold it is neither, and the size it does take — the two-thirds power — is an exponent with no elementary derivation that a measurement finds immediately.

 One threshold narrowing, one staying wide. Probability curves for connectivity and for containing a triangle, plotted against the edge probability as a multiple of each property's own threshold, at several graph sizes. Probability

Sharp, or merely a threshold

Every monotone property of a random graph has a threshold. Some of them turn on over a range that shrinks relative to the threshold as the graph grows, and some do not — and which kind a property is turns out to be decided by whether it is about a local structure or about the whole graph.

 A landscape nobody is looking at, and every move goes downhill on it. The 8 states of a congestion game with 3 participants and two resources, ordered by Rosenthal's potential, with every improving unilateral move drawn as an arrow. Every arrow points downward. Applied

The landscape nobody is looking at

Letting participants move one at a time to whatever is currently better can cycle forever, and on a network of congestible roads it cannot. The reason is a single number attached to each state that falls by exactly what the mover saves.

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