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

• Published in 2009
• Added on 2024-12-02

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.

Links

• http://arxiv.org/abs/0906.4747v1
• http://arxiv.org/pdf/0906.4747v1

Other information

BibTeX entry

@article{NonexistenceofPleatedFoldsHowPaperFoldsBetweenCreases,
	key = {NonexistenceofPleatedFoldsHowPaperFoldsBetweenCreases},
	type = {article},
	title = {(Non)existence of Pleated Folds: How Paper Folds Between Creases},
	author = {Erik D. Demaine and Martin L. Demaine and Vi Hart and Gregory N. Price and Tomohiro Tachi},
	abstract = {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.},
	comment = {},
	date_added = {2024-12-02},
	date_published = {2009-12-07},
	urls = {http://arxiv.org/abs/0906.4747v1,http://arxiv.org/pdf/0906.4747v1},
	collections = {drama,things-to-make-and-do},
	url = {http://arxiv.org/abs/0906.4747v1 http://arxiv.org/pdf/0906.4747v1},
	year = 2009,
	urldate = {2024-12-02},
	archivePrefix = {arXiv},
	eprint = {0906.4747},
	primaryClass = {cs.CG}
}