Celtic knots

Bands that dive under one another, never end and curl into tight loops are the signature of early medieval Insular art: the Lindisfarne Gospels, the Book of Kells (around 800), the stone crosses of Ireland and Scotland. Such a knot looks drawn freehand, but it is built on a grid, almost like cross-stitch.

How it works

It starts from a grid of dots. The bands run diagonally between them and cross at the midpoints of the grid edges, strictly in turn: over, under, over. With nothing in the way the result is a plain plait. Knots come from “Breaks”: a wall along a grid edge removes that crossing, and instead of an X the two bands turn back in smooth bends. The “Border” is a wall all the way round, closing the knot into a panel.

The parity rule makes the alternation exact: moving diagonally, a band changes the parity of its column at every crossing, and with it whether it lies on top. In topological terms every such knot is an alternating diagram. The last line is a small billiard theorem: inside a rectangular border a band travels diagonally like a ball off the cushions, so a 6 × 6 panel is six separate loops and a 5 × 3 panel a single strand.

A bit of history

Roman mosaicists already knew the plait, but it was Insular craftsmen of the 7th to 9th centuries who learned to break it into knots. Around 1900 John Romilly Allen showed that the knots are plaits with breaks. In 1951 the Scottish art teacher George Bain published Celtic Art: The Methods of Construction, which launched the revival of Celtic ornament. The French mathematician Christian Mercat set out this dots-and-walls method in detail, linking it to graph theory.

What to tweak

• “Breaks” at 0 with the “Border” on and “Colouring” “by strand”: the number of colours is the gcd of the panel's sides in cells. Add breaks and strands merge and split.
• “Break symmetry” “four-way” makes a carpet page like a manuscript's; “none” gives a free, asymmetric knot.
• Turn the “Border” off or switch to seamless tile mode and the plait becomes an endless strip or background.
• “Band width”, “Gap” and “Outline” range from thin cord to broad, crisply edged ribbons.
• The layer is vector, so SVG suits a plotter or a laser cutter; the “Relief” effect on top turns the knot into carved stone.