To make this process fairly automatic it's useful to put your motif "in a box", to encapsulate it, i.e. to draw a polygon around the interlacing. Mathematically oriented readers will already have worked out that it is then more natural to consider the dual graph from which you have constructed your interlacing: take a pencil of another colour, enclose your motif, surrounded by some space of an average half edge, inside a polygon. Consider an edge of your original graph, close to the polygon. Draw a line cutting this edge transversely, from the polygon to the centre of the face on the other side of the edge. Continue bit by bit, drawing a line for every edge of the graph. Once all your edges are cut by a line, the set of all these lines form the dual graph of your original graph. Don't hesitate to put several vertices on the same face if it's extended, you will join them into one by making them into a longitudinal wall. At the same time,if an edge of the original graph has a longitudinal block, the dual edge should have a transverse block and vice versa. The encapsulation of the preceding motif, built from a triangular grid, has for its dual a graph based on an hexagonal grid and enclosed in a continuous enclosure of walls:
It is important to know how to juggle between a graph and its dual; one of them is often really more simple than the other; so before learning the graph of your favourite motif by heart, look at its dual to see ... The key to encapsulation of interlacings is therefore to construct the dual graph which is associated with the one (more natural a priori) which had served to conceive it. You now have a graph for which all the external edges are longitudinal walls, it is enclosed in a polygonal rampart. And it's precisely this which allows you to easily elaborate more complex motifs by concatenation and juxtaposition of more simple encapsulated motifs. You just need to consider the two polygons put at the same scale, to align some external edges and to bring them close enough to make some of the walls of the two ramparts to coincide. You can then open one or two doors in the ramparts and the two interlacings will mix themselves together. And there's no need to find the dual graph - that's already encapsulated. Our triangular motif can then serve just as it is to construct a five branched star; you just need to deform the triangle a little and the graph which it contains then open a door so the knots join together:
The number of external edges that you join together don't necessarily have to be the same from one part to the other of the rampart; the external vertices of the two capsules don't have to coincide if you don't want to open a door at this point. Remember that to make a beautiful illumination you mustn't put the cart before the horse; don't try to create from scratch your intricate masterpiece on a large area, much better to take the time to construct an original freehand motif with which you are really happy and then to construct a large fraction of a total illumination, repenting at leisure at this motif. For this you must encapsulate it and pave the workspace with the form of its box. Then you recopy the graph of the motif inside each of the boxes, complete the exterior with a graph without flourishes and then open the corners of the boxes so the motif can escape and mix it all together into a recognizable extent. With this summary of the second method you are ready for the elaboration of a big frieze.
When you want to make a true illumination for a text or to enclose an object, carefully define the zone where you want your interlacing to be inscribed, for example a disk, a rectangle, a cross, a band etc. Divide the interior space into sub-zones according to taste and following the form which you want to create. The trace of the contour of these zones will be composed mostly of walls, being the edges of the graph that we are in the process of constructing, supporting a longitudinal block and forming a rampart. You had best be certain that you've chosen the scale of the design, i.e. the width of the ribbons you want to use. This width will determine the average length of the edges which should be roughly one and a half times bigger than the ribbons. Furnished with this length, place points periodically on the trace of the contour of the zones. These are in fact half-points laid down on one side only of the rampart, not extended from one part to another of the line. At the approach to a corner sharper than 90deg, increase the separation up to twice the normal if the angle is very acute. You'll notice that there must be an equilibrium between the length of the edges and the form of your zones. This trace should be made in grey as these are walls. You must open some doors if the sub-zones are disconnected; normalize some edges so that the zones will communicate between themselves. On the other hand, the perimeter must remain closed. When adding the vertices, make a graph grow harmoniously, its edges being approximately of the size you decided upon, vertices well spaced from each other, till the interior space is filled. To place the vertices well and to check the length of the edges, consider the edges like springs, the internal points to be mobile and relax them all whilst keeping certain points fixed. And voilà, your work is there awaiting only for you to make it happen. Arm yourself with patience and perseverence and the strands of your interlacing will emerge. Once you've worked-out the route of each thread (some hours later), you must expand them so that they fill out the walls and align themselves one against the other. Don't hesitate in this last phase to forget the graph and let your pencil express itself without constraint.
Christian Mercat 