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.
Let’s make one, partly based on Bain’s 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 8×6 here, and you’ll notice that I have accented every other line, so that there is a 4×3 grid of 2×2 cells.
Sketch diagonal lines so that each 2×2 cell has a rotated square in it:
Now, we are going to draw the border of our knot. Rotated squares on the corners will have three sides (shown in green) replaced. Rotated squares on the corners have two sides (shown in red) replaced.
Green sides are replaced with a cusp, and red sides with an arc.
It’s starting to look a bit like a Celtic knot. In fact, we could skip this next step, and produce a generic knot. But part of the art of creating a Celtic knot is varying its structure. Pick a number of interior intersections to be erased. I’ve erased two here.
After erasing each intersection, reconnect the strands vertically or horizontally. In my drawing, I’ve chosen to reconnect the broken lower left intersection with vertical lines
Next, we identify where the line passes over and under itself. Pick any intersection to start, and select which line goes under. Erase a little bit on each side of the overpassing knot. Then follow either line to the next intersection, and make sure the line does the opposite of what it did at the previous intersection. In other words, if it went over first, now it goes under. And vice versa.
Here is one of the beautiful results of such diagrams: alternating over and under always works. You cannot find a knot that cannot be completed this way. If you follow the paths, you will see that it is actually two knots that are connected. Mathematicians call this a link.
Finally, widen the lines, and you have created a Celtic knot!
We created tiles that allow you to create knots. If you look at each of the small blue squares in the knot above, you notice there are only a few types. These became our tiles. To give them more of a Celtic flavor, we added a rope motif to each tile.