onstructing a Celtic Knot

nitial Comments

While reading Iain's book, I spent several weeks drawing various knots, trying to get a feel for the method. I seemed to be spending a lot of that time in the initial construction phases, building the diagonal grid on top of the square grid. So much so that I started using graph paper, and even created some special graph paper for it. So I started looking for patterns, to find a way to simplify the construction.

This method was designed with the thought of writing a computer program to generate the knots. Therefore, I tried to come up with simple, deterministic rules, rather than rules which would require an artistic eye or complex judgement calls. In the details below, a construction is described for a simplified single cord design, rather than the more pleasing single cord or the visually complex double cord . However, I think that this technique could be extended to those (and other) styles of Celtic knotwork.

Also please note that this is not a method for constructing a work of Great Art; it is merely a mechanical approximation. The "final" knot lines from this method are close to where they should be, but sometimes need to be moved slightly, or smoothed out, for purely aesthetic reasons. This method is a first approximation only, and, as such, will be most pleasing when used to build large knots (large in number of crossings) where the overall pattern is more obvious than the execution.

onstructing a Knot

The colors used in this sample construction are described in the key to the right. Blueish lines are construction lines which will be erased at the end. Reddish lines are the actual knot, and will be (with small exceptions) permanent.

Step 1: Grid

Start with a grid with an even number of squares in both directions. Place dots at alternating vertices, making sure to miss the corners. To put it more technically, construct a (2m)x(2n) grid, and place dots at each (x,y) where x+y is odd.

The dots indicate where the "splittings" can be placed in the next two steps.

Step 2: Outside Borders

A splitting is a line through which the knot is not allowed to pass. These lines are drawn slightly shorter than two grid units long, and are centered on the dots drawn in the previous step.

Place splittings on all the dots along the outside border of the grid. These lines aren't actually splitting the knot, but the knot acts just the same when approaching an outside or an inside border, so they're useful during the knot drawing steps.

Step 3: Inside Borders

Now the inner splittings should be chosen. These splittings determine the eventual form of the knot, and should be chosen with care. Symmetric patterns are usually more visually pleasing, but I've chosen fairly odd splittings to demonstrate all the various curve forms. It may take a great deal of experimentation to get a feel for how the knot will look for a given set of splittings.

There are several guidelines to follow when deciding on internal splittings:

• All splittings are either horizontal or vertical.
• Only one splitting may be placed at each dot.

• If three splittings form a 'U' around a non-dot, there will be a "tailoff" (small loop) in the knot. These can add a certain flavor to a knot, but were discouraged by some knot makers.
• If a section of the grid is closed off (surrounded by splittings), an internal hole formed, and can be left blank. If it is filled in by a knot there will be two completely disconnected (not even interwoven) knots.
• Note that if no internal splittings are placed, the knot will be a simple gridwork.

Step 4: Straight Segments

This is the first actual drawing step; the first three only provided useful construction lines. At this stage, all remaining dots represent places where the knot will intersect itself.

Play "connect-the-dots" with adjacent dots (here, adjacent means diagonally adjacent; the sum of the absolute values of the differences in their coordinates is one).

Don't connect the dots completely; stop just before reaching each dot. Remember that these dots are the crossings, and only one of the knot segments is drawn at each crossing (the other one goes under). Crossings are dealt with in a later step.

Step 5: Short Curves

Going diagonally outwards from each dot, if the dot is approaching a splitting, draw a short curve out from just past the dot to the edge of the grid square. The curve should go straight towards the dot, and should be parallel to the splitting where it meets the grid edge.

To be more precise, the curve is an eighth of a circle of radius (the grid's unit length times the square root of two), centered at a nearby dot. If two short curves meet, they can be drawn at the same time (an entire fourth of a circle).

Once again, note that the curve should not quite reach the dot.

Step 6: Corners

Wherever a horizontal splitting and a vertical splitting meet in a corner, the knot will also have a sharp corner. When filling in these corners, draw at the same level as the short curves are at, so they meet smoothly. Since short curve ends are (square root of two) units away from a grid point, the lines of the corner piece should be (one minus the square root of two) units from the splittings.

Step 7: Long Line Pieces

Wherever there are thin stretches bounded by splittings all going in the same direction, the knot will be basically flat. These should be filled in with straight line segments.

Once again, the position of the segment should be such that it lines up with the short curves and the corners (not centered!). However, since long segments will never occur except next to short curves or corners, this should pose no problem.

Step 8: Centers

This step could actually have come before any of the drawing steps. However, it makes the diagrams appear more complicated. Do this step whenever it feels right.

Wherever there is a dot, fill in the center as follows:

• On odd rows, connect the lower right piece to the upper left piece.
• On even rows, connect the lower left piece to the upper right piece.

For a slightly different knot, swap the rules around. It makes little difference as long as consistency is maintained.

This is the step that would be the most difficult if the knot were drawn freehand. One would have to follow the knot around, alternating between going "over" and "under". That's the beauty of the mechanical construction; what once seemed magical and tricky becomes amazingly simple.

Step 9: Long Curves (optional)

This is the most complex step in all the constructions I've seen. Usually it is listed as optional, since doesn't affect the form of the knot, but merely makes it look much nicer.

A short curve followed by a long line segment looks a little odd, since it goes from very curved to completely flat quite suddenly. The two grid squares they occupy can be replaced by a single long, smooth curve.

This curve is also part of a circle (two long curves meet to form a smooth quarter of a circle) with a center several grid units away. But be warned that if it is drawn as a circle, the rest of the long line segment (and perhaps a corner) will have to be moved up slightly.

The true master knot constructor will see the long curves coming and draw them instead of the short curve/long line segment pair, rather than going back at the end and redrawing.

Step 10: Erase Construction Lines

Erase all construction lines.

Now the knot is complete!

Take the knot out and impress your friends with your amazing skill.

inal Comments

The drawing steps (four through nine) can come in any order. Drawing the centers first (step eight) can make it much easier to connect the pieces (and to keep from drawing the "under" pieces too close to the dots).

There is no reason why the initial grid has to be square, or even rectangular. As long as the width of any section of the grid shape is an even number of grid units, it will enclose an area that a knot can be drawn in.

In the strictest sense, every attempt should be made to make the Celtic knot be all one long piece (rather than several distinct, interwoven knots). However, it is difficult to see from the splittings whether or not the knot will all be in one piece. I usually make a quick sketch of the knot, count the number of components, and add or remove splittings to make it all one components before drawing the final knot -- hopefully without greatly altering whatever symmetries the knot was supposed to have.

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