Folds

First folding

From time to time I puzzle about the interesting ways in which cloth can fold, a puzzle noticed at reference 1. This morning was one of those times - and being only half awake it took rather a long time to come up with what seemed like promising ways into the problem.

Three examples. First, heavy curtains fold readily on a vertical axis. See, for example the illustration at reference 2. Second, hardboard, while reasonably flexible in large sheets, will become surprisingly strong and rigid if bent into a cylinder. Third, inflatables, tents and umbrellas do not fold flat without a good deal of often messy folding.

An important property of the cloth we have in mind is that it does not stretch or compress, or at least not very much. Suppose we mark a square grid on our piece of cloth while it is flat, with squares with sides of around 5mm. If then we fold that piece of cloth up, any old how, perhaps even scrunch it up loosely between our hands, most of those squares will still be more or less squares and the area of the triangle defined by any three of the four corners of a square will, for most of them, will be the same as it was, perhaps a little less but certainly no more.

First folding

It has proved, without bothering to make up a model to draw from, difficult to come up with a diagram which adds value. But maybe this one will do.

We have three poles of equal length, AB, CB and DB, flexibly joined at B, with A and C supported on a wall or something, B held up above both and D hanging down between them. We suppose that the triangle ACB only makes a modest angle with the horizontal, say less than 30 degrees, out of a possible 90.

But we say nothing about the ratio of AC to AD or, equally, that of AC to CD. AC might be quite small relative to AD, or it might be quite close to (but less than) AD + CD.

This framework is partially covered with a single piece of cloth; that is to say covering ADB and CDB, folded along DB and leaving ACB open.

One side of the cloth is shown as being of a darker colour than the other.

This piece of cloth can be unfolded to make a flat quadrilateral made up of two identical isosceles triangles, joined long side to long side.

Second folding

We then think of a square of cloth with four poles running from the corners to the centre, rather in the way of a simple tent. The loose ends of the poles are tied to the corners of the square of cloth. We then gradually pull the centre up, with the four poles. Gradually, the square marked out by the loose ends of the poles will get smaller - but what about the cloth?

In order to keep things simple, we suppose that there is nothing underneath the tent, no ground for anything other than the loose ends of the poles to rest on. It is not difficult to imagine a contraption which meets these requirements. It might, for example, involve rails pushing in towards the centre from the corners, over the hole, for the loose ends to rest on.

Quite quickly, we will find four folds of the first folding sort, hanging neatly down between each successive pair of poles. All that is missing, is the intermediate poles exemplified by DB in that first folding. In fact, superfluous, as gravity suffices. But maybe a round, rather than a sharp bend, out of respect for the constitution of the cloth.

But as the angle with the horizontal gets large, things start to break down and the folds no longer hang neatly. Both the space and the forces of gravity have become too small.

Third folding

For our third and last folding, we still have our large square of cloth, this time flat on a table. A table with a small hole in the middle, through which we slowly poke a long stick, rounded at the rising end so as not to pierce the cloth as it lifts it.

Quite quickly, we will find folds forming, although this time it is hard to be sure how many folds there are going to be and how even they are going to be, although, presumably, the flexibility and uniformity of the cloth will be factors here.

But quite quickly also, we will find things going wrong. In the first folding, point D is below points A and C, some of the cloth is falling down below the reference plane, as it were. In this case, the reference plane is a table and the cloth cannot fall down below, and so has to bunch up somehow. There have to be subsidiary folds. Which no doubt give rise to subsidiaries of their own, and so on.

If the cloth were more or less infinitely flexible, one might imagine a very large number of very small folds radiating from the top of our pole, folds which serve to get the cloth down from its starting diameter down to the diameter which is needed now. From where I associate to fractals.

Conclusions

No doubt all this can be generalised to the case noticed at reference 1, with the end point being general purpose algorithms to calculate the probability distribution of the configurations of cloth you are likely to get for any particular configuration of frame or support. Probability distribution because as paid-up Bayesians we always allow for a bit of random.

PS: all quite closely related to the equally tricky business of projecting parts of the surface of the earth onto the flat pages of an atlas.

Reference 1: http://psmv2.blogspot.com/2015/09/abstract-expressionism.html.

Reference 2: http://psmv3.blogspot.com/2018/02/on-counting-variations.html.

Reference 3: http://psmv2.blogspot.com/2014/09/botanic-problem-3.html. This problem might be topological too, but probably no relation to the present problem.