The tile
Paris, 1704. Sébastien Truchet — Carmelite father, typographer, canal engineer — publishes a memoir on a humble ceramic: a square split by its diagonal into two colours. Rotated through its four positions and set in rows, the one square yields, in his words, an infinity of designs.
The Méthode pour faire une infinité de desseins différents ran to seventy-two engraved plates. He needed exactly one tile to fill them.
The turn
In 1987 the metallurgist Cyril Stanley Smith redrew Truchet's diagonal as two quarter-circle arcs, each joining the midpoints of adjacent edges. The change is small and total: because every arc lands on a midpoint, any tile meets any neighbour in any rotation — flush, edge to edge.
Corridors appear that nobody drew. Turn one tile a quarter and a path across the whole field reroutes. That is what the waves above are doing, over and over, forever.
Three scales
This field runs Truchet at three nested scales — full, half and quarter tiles — after Christopher Carlson's multi-scale construction (Bridges, 2018). Halve a tile and its band halves with it; a parent's quarter-ring lands exactly where a child's band arrives, ports at ¼ and ½ of every edge, widths in strict octaves.
Where a full band meets a subdivided edge it ends in a round socket; where a child band meets a parent, the parent's corner ring receives it. Every port is checked at build time — position and width, to a tenth of a pixel.
— joints verified · — mismatches
The wave
Every dozen seconds a front crosses the field at twenty-seven degrees. Tiles in its wake quarter-turn — a little over half of them — and whole path systems reconnect: corridors fuse and take the orange, loops break, new ones close quietly behind the sweep. The grout never cracks, because a midpoint is a midpoint in every rotation.
The loop
When a corridor bites its own tail, the field notices. The tracer picks one closed loop, runs it in teal at three-fifths band width, holds it a beat, and lets go. Some loops close in four tiles; the long ones wander across forty before they find their own beginning.