Make a planar graph with signed edges with relatively few, let's say six or eight vertices and a dozen or so edges, sprinkled with signs (the same signs everywhere is often the prettiest), with longitudinal and transverse walls. Sharpen your pencil and go to it. You'll usually find the result, although relatively satisfying, lacking harmony and, in particular, if the underlying graph contains edges too different in scale, there will be big empty areas and places too full and confused where the threads are terribly crowded one against the other. And, more than anything, you have no control over the final product; you are even astonished that it looks like that (aren't you?). Let's begin at the beginning :
The simplest way to have a balanced graph is to start with a standard grid, square or triangular and only to sprinkle the blocks "appropriately". Appropriate usually means suitably symmetric or periodic. Here's an example of a triangular graph.
It is associated with the following knot :
The most rewarding for beginners is to get stuck in to some borders and the simplest are squares. Draw a dozen or so aligned squares, like a ladder or railway line. Distribute two or three walls and repeat them as you wish, making a series of reflections and translations about a central symmetry. You can't fail - it's superb.
Now try a series of two superimposed squares or a border based on triangles or on hexagons. You can subsequently bend your borders and make them into a circle; if its radius isn't too small then the motif isn't too deformed. On the other hand, for circles starting at the centre and spreading out to twenty times the grid, there's a problem of how to reduce it. You can't simply keep the sectors when you reduce the radius, you have to forget some of them. You should cleanly separate the different zones within which the radius doesn't change much by either putting in walls around most of a circumference or by joining some edges as in this example:
That's more or less the technique of the Irish monks. That already allows a very big diversity. But we mustn't stop there!
1. Towards the exterior
2. Towards the interior
The first one is to create small motifs more or less by freehand. The link between the two methods is encapsulation, once you have a motif that suits you, you can enclose it into a shape, a capsule. The second method is when you have a collection of motifs already encapsulated in a given shape, to create a whole page design with them, you first arrange on the page the shapes of the capsules as you wish, repeating them in patterns, e.g. putting them on a circle ... then you you draw inside each capsule the graph associated to it, you fill the rest of the space with some simple graph, you open some walls to let the tangles flow and that's it, your page looks like what you expected.
I suggest this method for creating small freehand motifs. It is quicker and less restricting than the second method which produces a neat result with the design fitting together as a whole. But we'll see that they work together. To start you produce some little graph with relatively few elements, say six or seven vertices and a dozen edges. Try to divide up the vertices equally, for example by imagining that the edges are springs. Buckle down and make the interlacing emerge from this structure. It won't work first time, you'll need to adjust the positions of the vertices, the lengths of the edges, a subtle alchemy for which I don't know the recipe. When the curves of the threads are well arranged, judge your creation. If you don't like it, throw it away or modify it locally. If you like it, store it carefully somewhere, it will serve you later for a larger-scale creation. When you have a set of motifs that please you it is really rather easy to assemble them into a bigger structure. All you need to do is draw the two graphs on the same scale, close enough to each other at a distance equal to the average length of the edges, then to add some edges coming from a vertex of one graph to a vertex of the other. Don't add more than one, two or three extra edges or else the original motifs will become unrecognizable. You just need to join them enough to unite them without losing their individuality. It's like a good marriage: not too disunited, and not too dominated by one partner or the other!
Christian Mercat
