Truchet Diagonals
Corner to corner, and a lattice of switchbacks.
Clock, date, widget row and the two bottom controls, drawn where iOS draws them.
Previewing with 8% bleed, as exported. Seed truchetdiagonals-001.
Which pattern you are looking at. Tap the preview to move to the next one and round again; the seed and the palette come with you, so the same words draw the same idea four ways.
Columns across the canvas. Rows follow from the aspect ratio so cells stay square.
How many parallel chords cross each cell. The corner-to-corner line becomes a family spaced one cell width over the count, which is the only spacing that tiles: it puts every crossing at a multiple of itself along each edge, in both rotations, so every chord meets a partner across every edge. A family is 2n-1 chords, so this stops at six — past that one cell carries more than a dozen lines and the tiling reads as grey.
How Truchet Diagonals works
A Truchet tile is a square with an asymmetric mark on it, and a Truchet tiling is what you get when you fill a grid with copies of that square in random rotations. One tile, four rotations and a coin flip per cell produce paths that wander across the whole grid and look considered in a way that no part of the rule accounts for. Sébastien Truchet was cataloguing floor tiles in 1704; the arrangement outlives the tile.
This one connects corners rather than edge midpoints. Paths meet at the cell’s corners, which makes the tiling read as a lattice of switchbacks rather than as the loops the arcs give you — sharper, more woven, and more obviously a grid. At one division it is a maze of diagonal zigzags, and it is the tile set that survives being pushed hardest.
Divisions does something here that the arcs cannot quite manage. The single corner-to-corner line becomes a family of parallel chords spaced one cell width over the count — the only spacing that tiles, because it puts every crossing at a multiple of itself along each edge, and puts them there in both rotations. Where a fan only meets its neighbour when the two cells agree on a corner, every chord here finds its partner across every edge whichever way the cell beyond it happens to be turned. Past three or four the cells stop reading as cells at all and the grid becomes a woven field of chevrons and nested diamonds. The count stops at six: a family is 2n-1 chords, so six already crosses one cell with eleven lines, and past that the tiling reads as grey rather than as a pattern.
The extent of the family is not a free choice either, which is why there is no spread control here. Truncating it to the chords nearest the diagonal leaves a cell crossing its right edge near one corner and its left edge near the other, so two neighbours turned the same way miss each other entirely. The spacing that makes the family join is the spacing that fills the cell.
Colour comes from a field drifting across the canvas, and each chord is stroked with a gradient sampled along its own length rather than painted one flat colour. Sampling once per chord sounds sufficient and is not: a colour boundary could then only fall in the gap between chords, and since every chord runs at 45° the field’s contours snapped onto a lattice of parallel lines and came out as straight-edged facets. Grid density decides the rest — below about six columns you read individual tiles, above about twenty a woven texture.
