Recursive Truchet Arcs

Divide a square into four, recurse, and drop one random quarter-arc into every cell that is left.

Divide, recurse, mark

Subdivision.QuadSubdivide(
    new Vector2(0f, 0f), Size, iterations,
    (center, halfSize) => DrawMark(canvas, paint, center.X, center.Y, halfSize, rng));

A square is split into four quadrants; each quadrant is split again; and so on for Iterations levels. Nothing is drawn on the way down. Only the leaves — the cells that are never subdivided further — receive a mark, and every leaf receives exactly one.

At depth n there are 4ⁿ leaves, each 2⁻ⁿ of the original size. The total ink therefore stays roughly constant while the detail quadruples, which is why the piece reads as the same texture at every depth rather than as a growing or thinning one.

Step 01

No subdivision. One leaf, one mark, at the full size of the square. This is the whole rule with the recursion removed.

What the mark is

Direction chooses between five mark sets:

Direction	Marks	Chance
0	vertical or horizontal bar	1/2 each
1	one of the two diagonals	1/2 each
2	one of four quarter-circle arcs	1/4 each
3	one of four semicircle arcs	1/4 each
4	a filled dot	always

Mode 2 is the canonical one. The four arcs are quarter circles of radius 2s centred on the cell’s four corners, so each one enters the cell at the midpoint of one edge and leaves at the midpoint of an adjacent edge.

That is the Truchet condition. Because every arc meets its cell boundary at an edge midpoint, and every neighbour’s arc does too, arcs join across cell edges into continuous curves — and the curves are not stored anywhere. They are what the eye does with a hundred independent quarter-turns.

Step 02

One division. Four marks, and already two of them may be continuing into each other.

Step 03

Sixteen. Long curves have appeared, running through several cells each, and none of them was drawn as a curve.

Each depth is a fresh drawing

The stages here do not nest, and it is worth saying why. Subdividing a leaf does not add detail inside an existing mark — it deletes that leaf and replaces it with four new ones, each of which draws its own mark from the next values in the random sequence. So the mark that occupied a cell at depth two is simply gone at depth three.

The progression is therefore the same rule at five depths, not one drawing being refined. That is what recursive subdivision actually is; a depth-three drawing is not a depth-two drawing with more in it.

Step 04

Sixty-four marks.

Step 05

Two hundred and fifty-six. At this size the individual arcs are small enough that the eye stops resolving them and reads only the paths they form.

Eight renderings

The variants push past recolouring: each changes how the rule is applied, and some change the rule itself.

Rendering	Cells	Direction	What it does
Truchet	256	arcs	The canonical piece: neon cyan on black.
Reef	1024	arcs	Two passes layered — sixteen large, faint currents under a reef of fine threads.
Tangle	256	arcs	A four-colour palette: each of the four arc orientations draws in its own colour, laying the curve families bare.
Zigzag	1024	diagonals	Diagonals meet at cell corners, chaining into sharp zig-zags instead of smooth curves.
Weave	1024	bars	Axis bars in a two-colour palette: every vertical one thread, every horizontal another, woven into a plaid.
Scales	256	scallops	Semicircle marks bulge into corners instead of through them; ink on warm paper.
Stardust	1024	dots	Dots nudged off their grid points into a loose constellation of gold.
Window	256	arcs	The same arcs pulled in from the edge and framed — a window into the pattern.

The same Truchet mechanism produces a completely different surface each time: arcs join at edge midpoints into curves, diagonals join at corners into zig-zags, and the jittered dots join into nothing at all.

The museum plate

The gallery ends with the piece the way it would hang on a wall. MuseumFrame is a presentation layer, not a generative one: it takes any ISkiaArt piece, clears to a gallery wall, and draws a frame, a mat with a bevel rule, and the artwork contain-fitted into the opening behind them. The lead preset, Truchet, appears in the gallery twice — once as the raw plate, once framed — because the frame is a way of showing the work rather than a rendering of it, and none of the other variants gets a framed copy.

The proportions are fixed: the opening is 65% of the canvas, the mat 32 pixels and the frame 16 pixels around it, and the rest is wall with a soft drop shadow under the frame. The artwork is clipped at the mat edge, so the piece’s own margins stay inside the opening instead of bleeding onto the wall.

Determinism

The generator is Xoshiro256StarStar by way of SeededRandom, seeded from the preset, and consumed once per leaf in the subdivision’s own traversal order — a jittered preset consumes two more values per leaf, and a layered one runs a second seeded stream for its underlay. It is one of the project’s stable generators, so the same seed produces the same marks on any machine and any runtime.

Progression 05

Variant 08

Gallery 01