# Why do folded concentric circles and rectangles form a hyperbolic paraboloid?

Here is a "self-forming" origami that I made from folding concentric circles - it would also happen if I folded concentric rectangles. How can the fold shapes such a saddle-like geometry?

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• This question appears to be off-topic because it is about paper-folding/origami and not physics. – Kyle Kanos Apr 22 '14 at 18:45
• @KyleKanos Origami would tell you how to make it, not why it folds the way it does. The solution clearly lies in analyzing what internal forces and stresses in the paper are created by the folds and bending, which is pretty squarely in physics territory. – Robert Mastragostino Apr 22 '14 at 19:54
• I think this is actually quite interesting, although it appears to be getting a lot of close-votes to have it moved to MathematicsSE. My only objection is that MathematicsSE tends to let a lot of questions go unanswered, so the question might not get the attention it deserves at MathSE versus at PhysicsSE (although admittedly this is more of a math than a physics thing). – DumpsterDoofus Apr 22 '14 at 20:36
• @DumpsterDoofus I agree they tend to leave questions unanswered, but this is clearly about manifolds, shapes, etc. So it still should be migrated – Jim Apr 22 '14 at 23:45
• @Jim: Agreed, voting to move. – DumpsterDoofus Apr 22 '14 at 23:53

Here is a very quick explanation. I'm sure the situation is analyzed in much more depth in the references in MvG's answer.

By pleating you have reduced the distance $r$ from the center of the disk to the circumference, but the length of the circumference $c$ has remained the same. There isn't room for the length $c$ of paper along the boundary to fit in the $2\pi r < c$ amount of space a flat disk would have, so it has to buckle.

In fact your shape is analogous to a disk in hyperbolic geometry, whose circumference is greater than $2\pi r$, and so it curls up when embedded in Euclidean space. The same thing happens in nature in leaves, jellyfish tentacles, and dried fruit.

If you had created pleats along radial lines instead, you'd have $c<2\pi r$ and the paper would form a cone, which is a rough approximation of spherical geometry.

• Thanks a lot Rahul. In lettuce you have many curls.why are there only two in this case? – Arthur Mamou-Mani Apr 24 '14 at 7:09
• I don't have an conclusive answer, but I can think of two reasons: (1) Paper is a much stiffer material, so it tries to minimize the amount of deformation. A two-curl saddle is the "lowest-frequency" mode for a disk to bend out of the plane. (2) Lettuce probably has much more excess material, that is, $c \gg 2\pi r$, so it needs more than two curls. See also: crochet models of hyperbolic space. – Rahul Apr 24 '14 at 18:08

You might want to look at (Non)Existence of Pleated Folds: How Paper Folds Between Creases by Demaine, Demaine, Hart, Price and Tachi from 2011. It analyses several aspects of the objects you are asking about. In particular (and for lack of time just judging from the first few pages), it claims that the version made from concentric rectangles cannot actually exist in a strict mathematical way, at least not with only the intentional creases. Additional slight creases are needed to make this work. A round version apparently would fold without creases, so your photo looks like it might agree with mathematical models of paper folding.

The authors of that paper are also good pointers for further information. Erik Demaine in particular hosts a page on Hyperbolic Paraboloids and also wrote a paper on Curved Crease Origami. I haven't researched the others as thoroughly.