Truchet Tiles — GeoTHM

The 1704 idea at its plainest: one motif, four orientations, a coin per tile.

Truchet’s original

Sébastien Truchet, a Dominican priest and typographer, published Mémoire sur les combinaisons in 1704 after noticing decorated tiles at a château. His question was combinatorial: how many patterns can be made from one square tile divided along a diagonal, laid at four orientations?

This piece is that question with arcs instead of a diagonal. Every tile carries the same motif; only the orientation varies, and it is a coin.

Step 01

A fifth of the grid.

The joining condition

The motif meets each edge of its tile at the midpoint. Its neighbour’s motif meets the shared edge at the same point, whatever orientation either took — so every arc end meets another arc end, at every interior edge, without exception.

That single constraint is the whole of Truchet tiling. Without it there would be 225 separate marks; with it there is a small number of long continuous paths, and the piece has a structure nothing in the code describes.

Step 02

Two fifths.

Step 03

Three fifths.

Step 04

Four fifths. The orientations are independent per tile — nothing biases a tile towards agreeing with its neighbour — so any long run you find is coincidence.

Step 05

The full grid.

Six renderings

Rendering	Grid	Gap	Size	Stroke	Ink
Cyber Cyan	15×15	12	11	0.30	cyan on black
Plasma Magenta	18×18	12	11	0.25	magenta on near-black
Matrix Green	12×12	12	11	0.35	green on near-black
Solar Gold	11×11	18	11	0.40	amber on near-black
Ghost White	10×10	16	14	0.65	white on near-black
Void Indigo	20×20	9	9	0.20	indigo on near-black

Gap against Size determines the spacing between tiles and shapes the character of each rendering. At 12 and 11 (Cyber Cyan, Plasma Magenta, Matrix Green) a hairline gap remains between cells, so the arc joins are implied rather than drawn. At 9 and 9 (Void Indigo) the tiles touch exactly and arc ends meet with no gap — curves flow seamlessly across the field. At 18 and 11 (Solar Gold) the tiles are clearly separated, making each tile’s pair of arcs readable as an isolated motif rather than part of a continuous path. Ghost White uses a much heavier stroke weight with larger arcs on a coarser grid, shifting the piece toward a graphic, printmaking register.

Each rendering has its own seed, so the six are independent draws.

Determinism

The generator is seeded from the preset and consumed once per tile, column by column. The generator is System.Random, which the project’s determinism policy flags as not guaranteed stable across .NET versions.

Progression 05

Variant 07