(Non)Existence of Pleated Folds: How Paper Folds Between Creases
(Non)Existence of Pleated Folds: How Paper Folds Between Creases
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DOI：
10.1007/s00373-011-1025-2
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发表时间：
2011-05-01
影响因子：
0.7
通讯作者：
Tachi, Tomohiro
中科院分区：
文献类型：
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作者：
Demaine, Erik D.;Demaine, Martin L.;Tachi, Tomohiro
我们证明了自1927年以来我们所熟悉的折纸模型双曲抛物面折纸实际上不能用零厚度纸的标准数学模型中的标准折痕图来折叠。相比之下，我们展示了该模型可以用额外的折痕折叠，这表明真正的纸可以通过小的折痕“折叠”成这个模型。我们推测这个模型的圆形版本，仅仅由同心圆折痕组成，也可以在没有额外折痕的情况下折叠。我们的研究结果的核心是一个新的结构定理，它描述了无折痕的内在平面——折痕之间的纸张部分。微分几何有很多关于这些表面的局部行为的说法，当它们足够光滑时，例如，它们是扭转的。但是这个经典的结果在整个表面的情况下是错误的。我们的结构特征说明了整个故事，甚至适用于二阶导数不连续的表面。我们用我们的定理来证明纸如何折叠的基本性质，例如，纸上的直折痕通过折叠必须保持分段直（多边形）。
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.