# Why Celtic knots work

When starting to design your own Celtic knots, some people worry - why do they work? Why is it, however free-form or complicated the knot is, the unders and overs always work out? Why do you never get two unders or two overs together? (Unless you make a mistake...)

 Start with a gap. Look at an edge. Put an arrow pointing towards where it goes under another line. You will find that the arrows will follow each other. They may be travelling either clockwise or anti-clockwise. This works however many lines surround the gap. The example on the left has four lines as it is a regular Celtic knot. The one on the right is more free-form, with three lines, but the arrows still follow each other.
 A Celtic knot will have clockwise and anti-clockwise gaps next to each other. So you can colour them in. There are always four gaps arranged round a "junction" (where one line goes under another). This is because there are only two lines crossing, and the gaps happen between the lines. So these four gaps are clockwise, anti-clockwise, clockwise, anti-clockwise. So it works.

### Why an internal dead-end messes it up

To work, the arrows must follow each other round a gap. But if there's a dead-end, the arrows can't follow each other. If the dead-end has just come from under another line, both arrows will point towards the "under". If the dead-end has come from an "over", they will point away. But these arrows point towards (or away) from each other, not following.

External dead-ends are fine! They aren't part of gaps.

There are always two version of any given Celtic knot. Just change the clockwise to anti-clockwise.