Circling the square

A while ago, I found a dazzling tiling pattern in Dillingen/Donau, southern Germany.

The tiling solves the problem of how to fill a circle with parallel tiles. The problem is only solvable incrementally, and in approximation, and this particular solution is interesting, because it adds local irregularities to achieve a global regularity.

It seems plausible to me that the whole thing is inspired by textile – pleating, and, perhaps even more, the incremental adding of stiches in knitting.


A very interesting solution to that particular problem. I too was especially reminded of the incremental adding of stitches in knitting.

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How regular the irregularities were, from the images it would seem that there is a semi-regular distribution of the irregularities, which would be very interesting. How irregular do they have to be to really allow the circle to be filled?

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This is an excellent question. I wish my investigation had been a little more systematic. But I do remember that I was fascinated by the impression that the decision to split a row was entirely local, when it seemed to be necessary.

Marking the split points (two different stones may play a role) shows that it at least isn’t a simple pattern.


What I find really interesting about this way of solving the problem, is the number of differently dimensioned paving blocks this solution requires, and the apparent random-ness of their use, as this photo shows. The rows which divide into two, gradually expand as paving blocks of incrementally increasing width are used, until a decision is made to split the row. It would appear that the rows which don’t divide are of uniform width from start to finish of the row, and it also looks like there are differing numbers of these non-expanding rows between the rows which expand and divide. It must have been interesting and challenging to make.

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Yes so for the expanding tiles they have a set of increasing size up to double the smallest width (plus space for cement).

So I think this works in a similar way to the Shephard tone / Risset rhythm illusions in music, where you slowly increase pitch or speed, adding elements whenever the frequency doubles so it seems to go up forever.

But then what would the equivalent illusion be here? Maybe that it looks like a radial grid pattern, but it’s really made of tree structures?

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Yes, this is how it is done: they have two types of tiles, one that stays the same, and one set that has all increasing sizes. The impression that of a straight line of tiles, where you don’t notice the increase, and suddenly there is a branch into a straight and an increasing one, while everything still fits. Optical perspective makes it even more dazzling.

In a Shepard tone you have a constant impression of increase while knowing that there must have been a jump. And here you encounter jumps, while knowing that they somehow must fit in.