Concentric Circles

There are lots of ways to get a sheet of paper to fold up nicely. Here is one way that I would never have come up with myself.

It is based on a simple fold-pattern by David Huffman: a disk with concentric mountain-valley circles, a single radius cut, and a tiny donut-hole in the middle. Based on? No, it is the same fold-pattern, the one from among the Huffman photos shown by the Institute for Figuring. (Unfortunately it was mistakenly labeled a “Tower of Concentric Circles”— although (a) it is not a tower and (b) Huffman has another model that is.)


Some tentative remarks on this form recently appeared in Erik & Marty Demaine’s useful if rather partial "history of curved origami sculpture". Yet at least as of this writing (mid-March 2008), the Demaines neglect to mention the most striking feature of such concentric circle patterns--and the one Huffman, surely, was exploring: the fact that you can do THIS:






Nice, no?

The math here is pretty straightforward: think how if you have a disk with a radius cut, you can make a cone of it by tucking one cut edge under the other, and then sharpen the cone continuously by twirling the cut edge underneath. Now notice that the same reasoning applies also to all horizontal slices of the cone (=concentric circles on the disk), which can be made into mountain & valley folds. It’s a neat illustration of several kinds of symmetry, and a way of folding a sheet into a quite small shape that does not involve any straight folds. --How small? Mathematically you could go on forever winding the thing up, but physics as usual gets in the way--here in the form of the thickness of the paper, which causes the surfaces at some point to stick.

So far as I can imagine, this trick will work smoothly ONLY with concentric circles (though there is one other spiral form which almost works too, albeit with surfaces not exactly flush to each other). The mountain-valley pairs need not be equally spaced, but they do need to be circular and to have a common center. So, I will hazard the claim that this seems to be a means of compacting paper that is unique to sheets curved-folded by means of concentric circular folds.


Now, leaving Huffman aside, with this very same model you can also explore a different property of concentrics: the one that the Demaines are interested in, following work which as they say was pursued over the last century by Josef Albers and his students in the Bauhaus, and by Thoki Yenn and Kunohiko Kasahara in the origami world.

Notice that when the form is wound up--with more than, say, one quarter of the circumference tucked under itself--the ridges of the mountain & valley folds add to the stability of the disk, which is flat on average. That is the familiar corrugation effect coming into play, the same method that is used to give wavy plastic rooftops and corrugated cardboard their added stability.

When you unwind it though, a strange thing happens. The corrugations weaken---that is not itself surprising, as the mountains/valleys are growing shallower. But long before the disk becomes altogether flat, the surface will have LESS stability than a comparable disk without the mountain and valley folds does. In fact, the disk starts to look for any excuse to break out of the plane: it refuses to stay flat! By the time the disk is unwound completely, it will naturally assume a contorted, saddle-like shape.

Why is that? Notice that the mountain and valley folds are adding some springiness to the paper, pulling the edges, and indeed every part of the interior, closer to the center. That means that the circumference now has to occupy the same space as a smaller circle. It can’t do that while remaining in the plane, so it bulges out of it. The same reasoning applies to each of the smaller circles, so you get a nice uniform twisty shape that is likely to be saddle-like.

If you hold a sheet of paper taught in your hands, and then move your hands closer together, the sheet will also bulge from the plane, for the same reason. It has nowhere to go but up or down.

The Demaines elsewhere write that “We know almost nothing about curved creases,” and in this context state that "forms that we are just beginning to understand" can be made by various new permutations of the concentric-circle technique. Presumably these expressions of ignorance or humility aren't being made only on the authors' own behalf, but for all of MIT, perhaps for all of Computational Origami so far as they are aware, or for 'art and science as a whole'. But the math, at least, that governs transitions in surfaces that undergo differential expansion (which is what is forcing the curvature) has been known for some time, and the physics too has been investigated at some depth--among others by Eran Sharon at the Hebrew University of Jerusalem (for a general discussion see American Scientist, 2004). More to the point for my purposes, this sort of curling or bending behavior is not unique to concentric circles: ANY folds that squeeze the interior of a sheet faster than the exterior, are going to cause the remoter parts to warp and bulge from the plane. The curling phenomenon will happen with concentric circles that don’t share a common center, with ellipses, with spirals, with suitable non-parallel curves, curves that do and that don’t intersect, indeed with suitable straight folds too. The outcome is, I grant especially elegant and ‘pure’ with concentric circles, but unlike the phenomenon Huffman was exploring this is not a property unique to concentrics or even to curved folding as such.

The results--as nicely exhibited by the Demaines in their work in the show presently at MoMA--are interesting and visually striking nevertheless. So it remains to be seen how this sort of technique can be adapted for the purposes of an origami that is not only mathematically inspired, but also expressive.