Hitomezashi

Stitch patterns from two sequences of coin flips: dashed lines alternate on-off along every row and column, each line’s starting phase is one bit, and the stitches join into closed loops that tile the page.

What it is

Hitomezashi (“one-stitch”) is a style of sashiko, the Japanese running-stitch embroidery used since the Edo period to strengthen and mend workwear. On a grid, the whole design is determined by one bit per row and one per column — and the emergent structure is the point: the segments always close into loops, and the enclosed regions are exactly two-colourable, which makes every page a natural colouring design.

How to use it

Colour the regions with two colours (it always works — that is a theorem), or follow any single loop with a pen. The bias knob stretches the texture: balanced coin flips give long snaking regions, near-uniform phases collapse into a regular weave of small cells.

History

Sashiko needlework is documented from Japan’s Edo period (1603–1868); hitomezashi is its grid-aligned single-stitch family. The mathematical structure — phase sequences, loop closure, two-colourability — was popularised by Annalisa Crannell and Katherine Seaton’s work and a 2021 Numberphile episode, which is what turned a stitching pattern into a generative-art staple.

The implementation’s guarantees

• The alternation invariant is tested per line, and the emergent regions are measured, not assumed: the page’s metadata reports the region count and the smallest region from a flood fill over the actual segments.
• The bias knob’s effect is pinned by a test — in the direction the measurement showed, which was the opposite of the first guess: extreme bias gives more, smaller regions (a regular weave), not fewer.
• Colourability runs with the adult rules; the escalation lever is stroke weight, since the pattern itself is fixed by its sequences.
• Rating basis: none — it is a design. The honesty fields are the region statistics and the segment count.