WING III · AXIOMATA
Mathematics is the only science whose objects cannot be photographed, and the only one whose results do not expire — a proof from Alexandria is still a proof, which is more than physics can say about 1900. Nothing on these five stages is an illustration of a result. Every line is computed from a definition, and wherever a quantity is known exactly it is printed beside the one that was measured.
EXHIBIT I · THE DISTRIBUTION OF PRIMES
Write the integers outward in a square spiral and mark the primes. Diagonals appear, and they are not an artefact of the drawing: a square spiral's diagonals are quadratic sequences, and some quadratics are far richer in primes than chance permits. Euler's n² + n + 41 is prime for the first forty values of n, and it lies along one of those lines.
The strip below is the same fact at a different scale. Z(t) is a real function with exactly the zeros that ζ(s) has on the critical line, and those zeros are the error term in the count of primes below x. Riemann guessed in 1859 that every one of them sits on that line. Nobody has found one that does not, and nobody has proved there is none.
hover a cell for its integer · t walks the critical line
EXHIBIT II · PERIOD DOUBLING
Iterate x → r·x(1−x). Below r = 3 the sequence settles on one value; above it, on two; then four, then eight — and the gaps between those doublings shrink by a constant factor. Feigenbaum found it in 1975 on a pocket calculator: δ = 4.669201…
That constant is not a property of this map. It belongs to every map with a smooth quadratic maximum, which is why it turns up in dripping taps and in cardiac arrhythmia. The value in the readout is bisected out of the picture you are looking at. Beside it, the Lyapunov exponent, which changes sign exactly where the cascade ends.
drag a box to zoom · double-click to reset · hover for the cobweb
EXHIBIT III · NEAREST NEIGHBOURS
Every pixel is coloured by which site it is nearest to. That is the entire definition, and it describes cell tissue, forest canopy, mobile-phone coverage and the grains in cast metal. Change the metric and the cells change shape — under L∞ a boundary can stop being a line at all, which is why this is solved per pixel rather than by clipping polygons.
Lloyd's algorithm moves each site to the centre of mass of its own cell, and repeats. Nothing in it knows what a hexagon is. Hexagons are what minimising the variance of the distance produces, and the readout counts them as they arrive.
drag a site · click to add one · shift-click to remove
EXHIBIT IV · REWRITING SYSTEMS
A fern does not store a picture of a fern. It stores a rule for replacing one part with a smaller arrangement of parts, and applies that rule to its own output. Lindenmayer wrote the formalism down in 1968 while studying the growth of algae, and it turned out to describe most of botany.
The grammar here is editable, and the readout measures the box-counting dimension of whatever you type — the exponent relating the number of boxes of size ε needed to cover the figure to ε itself. Where the exact value is known it is printed beside the measurement. Where it is not, there is a dash. Expect the two to disagree: the dragon curve is known to have dimension exactly 2, and box counting reads it as about 1.76, because the curve's finest detail is the length of one segment and no box here is smaller than that. Measurement lagging theory is the normal condition.
edit the grammar · | separates alternatives, picked at random
EXHIBIT V · KNOT THEORY
A (p, q) torus knot wraps p times around the axis while passing q times through the hole. It is a genuine knot only when p and q share no common factor. When they do, it is not one curve at all — it falls apart into gcd(p, q) separate loops, linked but individually unknotted. Set p = 4 and q = 2 and watch it come apart.
The crossings you can count in the picture depend on where you stand. The crossing number does not: it is the minimum over every possible view, and for a torus knot it is min(p(q−1), q(p−1)). Both are in the readout, so you can turn the knot and watch one of them refuse to move. Drawn by the wireframe rasteriser in core.js — there is no 3-D library on this page.
drag to orbit · shift-scroll to zoom