Abstract
Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500’s [Dür77], but have not been studied extensively until recently. Over the past few years, there has been a surge of interest in these problems in discrete and computationsl geometry. This paper gives a brief survey of some of the recent work in this area, subdivided into three sections based on the type of object being folded: linkages, paper, or polyhedra. See also [O’R98] for a related survey from this conference two years ago.
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References
H.-K. Ahn, P. Bose, J. Czyzowicz, N. Hanusse, E. Kranakis, and P. Morin. Flipping your lid. Geombinatorics, 10(2):57–63, 2000.
E. M. Arkin, M. A. Bender, E. D. Demaine, M. L. Demaine, J. S. B. Mitchell, S. Sethia. and S.S. Skiena. When can you fold a map? Computing Research Repository cs.CG/0011026, Nov, 2000. http://www/aeXiv.org/abs/cs.CG/0011026.
O. Aichholzer, C. Cortés, E. D. Demaine, V. Sujmović, J. Erickson, H. Meijer, M. Overmars, B. Palop, S. Ramaswami, and G. T. Toussaint. Flipturning polygons. In Proc. Japan Conf. Discrete Comput. Geom., Lecture Notes in Comput. Sci., Tokyo, Japan, Nov. 2000. To appear in Descrete and Computational Geometry.
O. Aichholzer, E. D. Demaina, J. Erickson, F. Hurtado, M. Overmars, M. A. Soss, and G. T. Toussaint. Reconfiguring convex polygons. Comput. Geom. Theory Appl., 2001. To appear.
B. Aronov, J. E. Goodman, and R. Pollack. Convexification of planar polygons in R3. Manuscript, Oct. 1999. http://www.math.nyu.edu/faculty/pollack/convexifyingapolygon10-27-99.ps.
B. Aronov and J. O’Rourke. Nonoverlap of the star unfolding. Discrete Comput. Geom., 8(3):219–250, 1992.
T. Beidl, E. Demaine, M. Demaine, A. Lubiw, M. Overmars, J. O.’Rourke, S. Robbins, and S. Whitesides. Unfolding some classes of orthogonal polyhedra. In Proc. 10th Canadian Conf. Comput. Geom., Montréal, Canada, Aug. 1998. http://cgm.cs.mcgill.ca/cccg98/proceedings/cccg98-biedl-unfolding.ps.gz.
T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O’Rourke, M. Overmars, S. Robbins, I. Streinu, G. Toussaint, and S. Whitesides. Locked and unlocked polygonal chains in 3D. Technical Report 060, Smith College, 1999. A preliminary version appeared in the Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, Baltimore, Maryland, Jan. 1999, pages 866–867.
T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O’Rourke, S. Robbins, I. Streinu, G. Toussaint, and S. Whitesides. A note on reconfiguring tree linkages: Trees can lock. Discrete Appl. Math., 2001. To appear.
M. Bern, E. D. Demaine, D. Eppstein, E. Kuo, A. Mantler, and J. Snoeyink. Ununfoldable polyhedra with convex faces. Comput. Geom. Theory Appl. 2001. To appear.
M. Bern, E. Demaine, D. Eppstein, and B. Hayes, A disk-packing algorithm for an origami magic trick. In Proc. Internat. Conf. Fun with Algorithms, Isola d’Elba, Italy, June 1998.
T. C. Biedl, E. D. Demaine, S. Lazard, S. M. Robbins, and M. A. Soss. Convexifying monotone polygons. In Proc. Internat. Symp. Algorithms and Computation, volume 1741 of Lecture Notes in Comput. Sci., pages 415–424, Chennai, India, Dec. 1999.
M. Bern and B. Hayes. The complexity of flat origami. In Proc. 7th ACMSIAM Sympos. Discrete Algorithms, pages 175–183, Atlanta, Jan. 1996.
R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41st IEEE Sympos. Found. Comp. Sci., pages 432–442, Redondo Beach, California, Nov. 2000.
J. Cantarella and H. Johnston. Nontrivial embeddings of polygonal intervals and unknots in 3-space. J. Knot Theory Ramifications, 7(8):1027–1039, 1998.
J.A. Calvo, D. Krizanc, P. Morin, M. Soss, and G. Toussaint. Convexifying polygons with simple projections. Infor. Process. Lett., 2001. To appear.
R. Cocan and J. O’Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canadian Conf. Comput. Geom., Vancouver, Canada. Aug. 1999. http://www.cs.ubc.ca/conferences/CCCG/elec_proc/c17.ps.gz.
E. D. Demaine and M. L. Demaine. Recent results in computational origami. In Proc. 3rd Internat. Meeting of Origami Science, Math, and Education, Monterey, California, March 2001. To appear.
E. D. Demaine, M. L. Demaine, and A. Lubiw. Folding and cutting paper. In J. Akiyama, M. Kano, and M. Urabe, editors, Revised Papers from the Japan Conf. Discrete Comput. Geom., volume 1763 of Lecture Notes in Comput. Sci., pages 104–117, Tokyo, Japan, Dec. 1998.
E. D. Demaine, M. L. Demaine, and A. Lubiw. Flattening polyhedra. Manuscript, 2000.
E. Demaine, M. Demaine, A. Lubiw, and J. O’Rourke. Examples, counterexamples, and enumeration results for foldings and unfoldings between polygons and polytopes. Technical Report 069, Smith College, Northampton, MA, July 2000.
E. D. Demaine, M. L. Demaine, A. Lubiw, and J. O’Rourke. Enumerating foldings and unfoldings between polygons and polytopes. In Proc. Japan Conf. Discrete Comput. Geom., Lecture Notes in Comput. Sci., Tokyo, Japan, Nov. 2000.
E. D. Demaine, M. L. Demaine, and J. S. B. Mitchell. Folding flat silhouttes and wrapping polyhedral packages: New results in computational origami. Comput. Geom. Theory Appl., 16(1):3–21, 2000.
A. Dürer. The Painter’s Manual: A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler Assembled by Albrecht Dürer for the Use of All Lovers of Art with Appropriate Illustrations Arranged to be Printed in the Year MDXXV. Abaris Books, Inc., New York, 1977. English translation of Unterweysung der Messung mit dem Zirkel un Richtscheyt in Linien Ebnen uhnd Gantzen Corporen, 1525.
H. Everett, S. Lazard, S. Robbins, H. Schröder, and S. Whitesides. Convexifying star-shaped polygons. In Proc. 10th Canadian Conf. Comput. Geom., Montréal, Canada, Aug. 1998. http://cgm.cs.mcgill.ca/cccg98/proceedings/cccg98-everett-convexifying.ps.gz.
P. Erdös. Problem 3763. Amer. Math. Monthly, 42:627, 1935.
T. Fevens, A. Hernandez, A. Mesa, M. Soss, and G. Toussaint. Simple polygons that cannot be deflated. Beiträge Algebra Geom., 2001. To appear.
M. Gardner. The combinatorics of paper folding. In Wheels, Life and Other Mathematical Amusements, chapter 7, pages 60–73. W. H. Freeman and Company, 1983.
B. Günbaum. How to convexify a polygon. Geombinatorics, 5:24–30, July 1995.
T. Hull. On the mathematics of flat origamis. Congr. Numer., 100:215–224, 1994.
J. Justin. Towards a mathematical theory of origami. In K. Miura, editor, Proc. 2nd Internat. Meeting of Origami Science and Scientific Origami, pages 15–29, Otsu, Japan, November—December, 1994.
T. Kawasaki. On the relation betwen mountain-creases and valley-creases of a flat origami. In H. Huzita, editor, Proc. 1st Internat. Meeting of Origami Science and Technology, pages 229–237, Ferrara, Italy, Dec. 1989. An unabridged Japanese version appeared in Sasebo College of Technology Report, 27:153—157, 1990.
M. Kapovich and J. Millson. On the moduli space of polygons in the Euclidean plane. J. Differential Geom., 42(1):133–164, 1995.
R. J. Lang. A computational algorithm for origami design. In Proc. 12th Sympos. Comput. Geom., pages 98–105, Philadelphia, PA, May 1996.
A. Lubiw and J. O’Rourke. When can a polygon fold to a polytope? Technical Report 048, Smith College, June 1996.
W. F. Lunnon. Multi-dimensional map-folding. The Computer Journal, 14(1):75–80, Feb. 1971.
W. J. Lenhart and S. H. Whitesides. Reconfiguring closed polygonal chaines in Euclidean d-space. Discrete Comput. Geom., 13:123–140, 1995.
J. S. B. Mitchell, D. M. Mount, and C. H. Papadimitriou. The discrete geodesic problem. SIAM J. Comput., 16(4):644–668, Aug. 1987.
J. Montroll. African Animals in Origami. Dover Publications, 1991.
B. Nagy. Solution to problem 3763. Amer. Math. Monthly, 46:176–177, Mar. 1939.
J. O’Rourke. Folding and unfolding in computational geometry. In Revised Papers from the Japan Conf. Discrete Comput. Geom., volume 1763 of Lecture Notes in Comput. Sci., pages 258–266, Tokyo, Japan, Dec. 1998.
G. T. Sallee. Stretching chords of space curves. Geom. Dedicata, 2:311–315, 1973.
C. Schevon. Algorithms for Geodesics on Polytopes. PhD thesis, Johns Hopkins University, 1989.
G. C. Shephard. Convex polytopes with convex nets. Math. Proc. Cambridge Philos. Soc., 78:389–403, 1975.
I. Streinu. A combinatorial approach to planar non-colliding robot arm motion planning. In Proc. 41st IEEE Sympos. Found. Comp. Sci., pages 443–453, Redondo Beach, California, Nov. 2000.
G. Toussaint. Computational polygonal entanglement theory. In Proceedings of the VIII Encuentros de Geometria Computational. Castellon, Spain, July 1999.
G. Toussaint. The Erdös-Nagy theorem and its ramifications. In Proc. 11th Canadian Conf. Comput. Geom., Vancouver, Canada, Aug. 1999. http://www.cs.ubc.ca/conferences/CCCG/elec_proc/fp19.ps.gz.
G. Toussaint. A New class of stuck unknots in pol 6. Beiträge Algebra Geom., 2001. To appear.
S. Whitesides. Algorithmic issues in the geometry of planar linkage movement. Australian Computer Journal, 24(2):42–50, May 1992.
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Demaine, E.D. (2001). Folding and Unfolding Linkages, Paper, and Polyhedra. In: Akiyama, J., Kano, M., Urabe, M. (eds) Discrete and Computational Geometry. JCDCG 2000. Lecture Notes in Computer Science, vol 2098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47738-1_9
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