Celtic Plait

An interlaced plait: cords crossing over and under, turned back by chosen cells, every cord provably a closed loop.

What it is

Cords run through the midpoints of cell edges. Each cell either crosses its two strands (one passing visibly over the other) or turns them back; the border joins the outer midpoints in pairs. The over/under alternates along every cord, and the gaps at each under-crossing are what make the page read as woven rather than as crossing lines.

What it draws is a plait — cords along the rows and columns. The 45° diagonal interlace of Insular manuscript borders needs a different construction (cords matched under rotation) and is recorded as not implemented rather than approximated.

How to use

Colouring and tracing pages. Tracing a single cord over and under until it returns to its start is the classic exercise; colouring by cord (each closed loop one colour) shows the knot’s structure — the cords count in the metadata says how many colours that takes.

Purpose

The knotwork entry in the design lane, built on a proof rather than a check: every midpoint acquires exactly two connections (one per adjacent cell, plus the border pairing), and a graph where every node has degree two is a disjoint union of cycles — so a dangling cord is not a reachable state. The crate documents its abandoned first model (a bouncing billiard) whose stepping was non-invertible and whose over/under rule was arithmetically impossible; property tests caught both.

History

Interlace patterns predate the Celts (Roman mosaics, Coptic manuscripts), but the Insular tradition of the 7th–9th centuries — the Book of Kells, the Lindisfarne Gospels — made knotwork its signature, woven into borders, initials and carpet pages. Modern construction methods (grids, breaklines) descend from George Bain’s Celtic Art: The Methods of Construction (1951).

This implementation

• Spec knobs: size, cells, break_share (fraction of cells that turn the cords), gap (the under-crossing window), stroke, kids.
• Generation: a grid of crossing cells with turns placed at random; degree-two connectivity checked (a failure is a model bug, reported as such); cords traced deterministically from sorted starts; dull knots (a mesh of short loops) rejected by measure and re-rolled.
• Guarantees: every cord closes — by counting argument, verified per page; over/under alternates along every cord (cell-parity rule, tested); deterministic per seed; colourability-gated with grid coarsening.