(Non)Existence of Pleated Folds: How Paper Folds Between Creases

We prove that the pleated hyperbolic paraboloid, a familiar origami model known since 1927, in fact cannot be folded with the standard crease pattern in the standard mathematical model of zero-thickness paper. In contrast, we show that the model can be folded with additional creases, suggesting that...

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Bibliographic Details
Published in:Graphs and combinatorics 2011-05, Vol.27 (3), p.377-397
Main Authors: Demaine, Erik D., Demaine, Martin L., Hart, Vi, Price, Gregory N., Tachi, Tomohiro
Format: Article
Language:English
Subjects:
Combinatorial analysis
Combinatorics
Computational mathematics
Derivatives
Discontinuity
Engineering Design
Geometry
Graphs
Mathematical models
Mathematics
Mathematics and Statistics
Proceedings Paper
Theorems
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Summary:We prove that the pleated hyperbolic paraboloid, a familiar origami model known since 1927, in fact cannot be folded with the standard crease pattern in the standard mathematical model of zero-thickness paper. In contrast, we show that the model can be folded with additional creases, suggesting that real paper “folds” into this model via small such creases. We conjecture that the circular version of this model, consisting simply of concentric circular creases, also folds without extra creases. At the heart of our results is a new structural theorem characterizing uncreased intrinsically flat surfaces—the portions of paper between the creases. Differential geometry has much to say about the local behavior of such surfaces when they are sufficiently smooth, e.g., that they are torsal ruled. But this classic result is simply false in the context of the whole surface. Our structural characterization tells the whole story, and even applies to surfaces with discontinuities in the second derivative. We use our theorem to prove fundamental properties about how paper folds, for example, that straight creases on the piece of paper must remain piecewise-straight (polygonal) by folding.
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-011-1025-2