Plates on patterns that are perfectly ordered and never repeat, each one computed in the browser as you watch: the Fibonacci chain, cut from a square lattice by a strip of slope 1/φ, grown by the rule that turns a long piece into a long and a short, repeating for every rational slope and never for this one; and Penrose's two rhombs, whose arcs must meet across every edge, cut again and again into smaller rhombs, with gold curves that close with fivefold symmetry and a pattern that splits and returns as you zoom in; and de Bruijn's pentagrid, five families of parallel lines whose crossings become the same rhombs, which flip in hexagons as the lines slide and turn into the squares and rhombs of the Ammann–Beenker tiling with four families at 45°; and the pinwheel, one right triangle cut into five copies of itself, whose tiles turn by arctan(1/2), no rational part of a turn, and so point in more and more directions until they fill the circle; and the hat, the single shape proved in 2023 to cover the plane only aperiodically, built of four clusters that grow into larger copies of themselves, with φ⁴ as many hats facing up as turned over, and one of a family of shapes, Tile(a, b), into which its tilings bend. Most plates keep in their corner a count of the tiles on screen, two kinds of them, and set their ratio against the irrational number it tends to, which no periodic pattern could have; the pinwheel counts instead the directions its tiles point in, which a periodic tiling could have only finitely many of.
図版
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各 60 秒
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再生エリアに合わせて自動調整（最大 440 万画素）
あらかじめ録画したコマ
なし
映画
制作中
図版 I · 全 4 楽章 · 1:00 のループ
The Fibonacci chain
The simplest aperiodic order lives on a line. Lay a strip across the square lattice at slope 1/φ, as wide as a unit square seen edge-on, and drop the lattice points inside it onto its centre line: steps to the right become long pieces and steps up short ones, in the golden ratio. The sequence never repeats, yet it is far from random: the rule that turns every long piece into a long and a short, and every short into a long, grows it; the ratio of the counts tends to φ; and every word that appears in it appears again and again, at one of exactly two distances. The corner counts the long and short pieces on screen and sets their ratio against φ.
図版
解説
The count in the corner
Most plates show two kinds of tile: long and short pieces, thick and thin rhombs, squares and rhombs, hats facing up and turned over. The corner counts both on screen and divides one count by the other. For a pattern that repeats, the ratio tends to the ratio in one period, a fraction; for these it tends to an irrational number, φ, √2 or φ⁴, and a small curve at the right plots it against the size of the patch, crossing and recrossing the limit as it closes in. An irrational ratio is one way to see that a pattern can never repeat. The pinwheel counts something else, the directions in which its tiles point, and they grow without end.
Order without repetition
Nothing in these patterns is random. Each is fixed down to the last tile by a rule, a strip across a lattice or a way of cutting tiles into smaller tiles, and each is repetitive in a strong sense: every finite piece of it appears again and again, within a bounded distance of anywhere. What they lack is a translation that carries the whole pattern onto itself.
Computed, not stored
The tiling of the second plate is not stored anywhere. For every pixel the shader starts from the ten triangles around the centre, picks the one the pixel is in, cuts it, and follows the piece that contains the pixel, a dozen times over; the edges and arcs it draws are those of the last piece. The counts in the corner are made the same way on the CPU, for the whole wheel or for the part on screen.