4 by 4 weaves
Consider the plain weave (4th position, bottom row). Over, under, over, under. Very boring. Or the basket weave (bottom right). Over, under, two at a time. If the basic unit of our pattern is 4 threads by 4 threads, how many different patterns are there? At each crossing, the horizontal thread can be over or under the vertical thread, so 2 to the 16th, or 65,536. I decided to add a couple of constraints:
- Each horizontal thread goes over twice and under twice
- Each vertical thread goes over twice and under twice
Together, these constraints result in a “balanced” pattern. For example, if the horizontal threads are white and the vertical threads are blue, each side of the cloth will show half white and half blue. Now we have a more interesting problem, and we can have the computer grind out all 65,536 patterns and count the overs and unders in each row and column. The we eliminate near duplicates where one pattern is like another except shifted over by one or two threads.
It turns out that there are exactly 10 weaves that satisfy our constraints, and they’re shown above. In addition to the plain weave and the basket weaves, we have a couple of 2/2 twills (last 2 in top row).
Alternatively, we can count patterns as being the same if one can be rotated or transposed into the other. For example, if we take one of our 2/2 twills and rotate it 90 degrees, we get the other 2/2 twill, so we could count them as the same pattern. If we count this way, we get 6 patterns instead of 10.
The most interesting patterns, I think, are the ones on the left. If I look at the top left pattern one way, it looks like diagonal white chains on a blue background. If I tilt my head a bit, I see blue chains on a white background, on the opposite diagonal. What can I say? I have astigmatism and I’m easily amused.