Most of the Celtic knots on this website are based on a square grid. The historic examples seem to prefer this, and I'm not surprised. It is quite easy to draw a square grid by eye, but other angles are far harder. Computers prefer square grids as well!
However, it is possible to have an under-and-over pattern based on a triangular grid (see right).
The first problem is that the three different lines meet at a point. How do we decide which line goes over or under which? What you need to do is to shift one of the set of lines, so you only have a pair of lines at each junction.
First, delete the horizontal lines.
Double these up to produce the strands.
Now put the horizontal lines back, shifting them up a little, and making the under-and-overs.
To finish the knot, tidy up the edges, and colour it in. Here I have removed the background as well.
One property of this type of grid is that it looks quite different if you turn it.
The previous example was a loose knot. You can tighten the knot a lot more, as in this example. On the left, I show how you can just colour in the grid to make the strands. On the right, I've removed the surplus grid lines and removed the background. I've made the different orientations as different colours, so I couldn't tidy up the edges.
This does high-light another disadvantage of this grid. There are large gaps between the strands, even with a tight knot.
Most regular tessellations apart from squares and triangles have an odd number of numbers at each junction, which makes it impossible to make into a celtic knot. However, there is a tessellation of squares, hexagons and triangles. Here there are four lines at each junction, unlike the triangles above, so it is easier to make a knot from it. It ends up an interlocking dodecagons!