Celtic knots

To artists and mathematicians, knots can be beautiful, interesting structures. Until recently, I hadn't explored them much from either perspective. Then my mother-in-law gave me a copy of George Bain's Celtic Art: The Methods of Construction for my birthday Bain shows how to create Celtic knots, from simple to elaborate.

Mathematicians require that a knot be closed, which means it has no loose ends. Celtic art from the Middle Ages is renowned for its intricate closed knots.

It is believed that Celtic artists began their works with a sketch of diagonal grid lines. You can create a design, inspired by Bain's interpretation of Celtic techniques. Celtic knots can be used to fill complex shapes, but many of the motifs fill rectangles, and this is a good place to start. You may want graph paper, pencil, and an eraser. Create a grid with an even length on each side. I'm using 8x6 here, and you'll notice that I have accented every other line, so that there is a 4x3 grid of 2x2 cells.

Now replace edges and selected intersections with arcs, and make the intersections weave over and under.

If you study each cell in the grid, you'll notice that there are only 6 kinds of cells, not counting rotations of the cells:

Now we transform these simple sketches into a more ornate drawing, reflecting the detail of Celtic art:

These tiles can be used to represent artwork found in Bain's book…

At this point, we were almost satisfied with our design. We decided to add one more tile that just goes straight from one side to the other:

Once you have created a design, it becomes apparent you can easily swap a few tiles to get a new design. Here's a demonstration of generating 300 random Celtic knots in ten seconds by making a succession of small changes:

And here is complete Knot puzzle kit: