Greek keys - triangular patterns

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Houelt cross

Greek keys are square. Everyone knows that!

I've recently visited Llantwit Church (near Cardiff), an ancient Christian site with some splendid Celtic crosses and carved stones in it. My main interest was the Celtic knots on the stones, but there was an attractive repeated pattern on tthe front of the Houelt Cross closely. I thought that I would copy it, and looking at it closer, discovered that it was a 2 dimensional triangular Greek key.

The pattern is on the stem of the cross, in the middle of the photo. The lines of keys run up and down, see right.

Here is one line, horizontal:

triangular Greek key

It doesn't seem right to call it a Greek key. Perhaps it should be a Celtic or Welsh key.

triangular Greek key on the Houelt Cross in Llantwit church

In reproducing this pattern, I've made the slanting lines at 45° as this produces a smoother line on a computer (because of pixellation). I think that 60° might make a more attractive pattern (see below).

triangular Greek key with 60 degrees

Triangular keys have several interesting developments. Firstly, if you look at the 2 dimensional pattern above, you can see that there are not four independent lines. The lefthand line joins to the next one along, and in fact it's a continuous line with no beginning or end. There are two of these on the cross, side by side. That is interesting in itself, but we can also construct a 2 dimensional pattern as big as we want it, which is one line. It does have a start and end, but it is a true 2 dimensional pattern. There is both a 45° and 60° version.

triangular 2 dimensional Greek key triangular 2 dimensional Greek key

Perhaps even more exiciting, we can use triangular Greek keys (Welsh keys!) to solve the problem of corners in a border. I have already discussed here the problem of corners in conventional Greek keys. Christopher Stanton has come up with some unconventional ideas. But if you use triangular keys then the solution is easy. This does need the 45° angle, and that means that you can join together two lines at the 45° edge. The corners are not quite the same; two alternate ones have a line at the outside and a gap on the inside, and the others are the opposite. We can disguise this by using the usual Greek key trick of a line outlining the border (see below). Of course, if you used 60° angles, then you could have a hexagon as a border!

Welsh key border

Welsh key border