Paper by Martin L. Demaine

Erik D. Demaine, Martin L. Demaine, and Joseph S. B. Mitchell, “Folding Flat Silhouettes and Wrapping Polyhedral Packages: New Results in Computational Origami”, Computational Geometry: Theory and Applications, volume 16, number 1, 2000, pages 3–21. Special issue of selected papers from the 3rd CGC Workshop on Computational Geometry, 1998.

We show a remarkable fact about folding paper: From a single rectangular sheet of paper, one can fold it into a flat origami that takes the (scaled) shape of any connected polygonal region, even if it has holes. This resolves a long-standing open problem in origami design. Our proof is constructive, utilizing tools of computational geometry, resulting in efficient algorithms for achieving the target silhouette.

We show further that if the paper has a different color on each side, we can form any connected polygonal pattern of two colors. Our results apply also to polyhedral surfaces, showing that any polyhedron can be “wrapped” by folding a strip of paper around it. We give three methods for solving these problems: the first uses a thin strip whose area is arbitrarily close to optimal; the second allows wider strips to be used; and the third varies the strip width to optimize the number or length of visible “seams” subject to some restrictions.

This video by Paula Beardell Krieg shows how to create some cool models with the color reversal gadget of this paper.

The paper is \copyright Elsevier Science Netherlands.

The paper is 23 pages.

The paper is available in PostScript (647k), gzipped PostScript (132k), and PDF (221k).
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Related papers:
SoCG99 (Folding Flat Silhouettes and Wrapping Polyhedral Packages: New Results in Computational Origami)
CGC98 (Folding Any Silhouette from a Strip)

Related webpages:
Wrapping Polyhedra (Erik Demaine)

See also other papers by Martin Demaine.
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Last updated November 17, 2022 by Martin Demaine.