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Notes and References Notes and References for Chapter 2 Page 34 a practical construction ... see Morey above, and also Jeremy Shafer, Page 14 limit yourself to using only one or “Icosahedron balloon ball,” Bay Area Rapid two different shapes . These types of starting Folders Newsletter, Summer/Fall 2007, and points were suggested by Adrian Pine. For many Rolf Eckhardt, Asif Karim, and Marcus more suggestions for polyhedra-making activi- Rehbein, “Balloon molecules,” http://www. ties, see Pinel’s 24-page booklet Mathematical balloonmolecules.com/, teaching molecular Activity Tiles Handbook, Association of Teachers models from balloons since 2000. First described of Mathematics, Unit 7 Prime Industrial Park, in Nachrichten aus der Chemie 48:1541–1542, Shaftesbury Street, Derby DEE23 8YB, United December 2000. The German website, http:// Kingdom website; www.atm.org.uk/. www.ballonmolekuele.de/, has more information Page 17 “What-If-Not Strategy” ... S. I. including actual constructions. Brown and M. I. Walter, The Art of Problem Page 37 family of computationally Posing, Hillsdale, N.J.; Lawrence Erlbaum, 1983. intractable “NP-complete” problems ... Page 18 easy to visualize. ... Marion Michael R. Garey and David S. Johnson, Walter, “On Constructing Deltahedra,” Wiskobas Computers and Intractability: A Guide to the Bulletin, Jaargang 5/6, I.O.W.O., Utrecht, Theory of NP-Completeness, W. H. Freeman & Aug. 1976. Co., 1979. Page 19 may be ordered ... from theAs- Page 37 the graph’s bloon number ... sociation of Teachers of Mathematics (address Erik D. Demaine, Martin L. Demaine, and above). Vi Hart. “Computational balloon twisting: The Page 28 hexagonal kaleidocycle ... theory of balloon polyhedra,” in Proceedings D. Schattschneider and W. Walker, M. C. Escher of the 20th Canadian Conference on Compu- Kaleidocycles, Pomegranate Communications, tational Geometry, Montreal,´ Canada, August Petaluma, CA; Taschen, Cologne, Germany. 2008. Page 33 we suggest you try it ... If Page 37 tetrahedron requires only two bal- you get stuck, see Jim Morey, YouTube: loons . see Eckhardt et al., above. theAmatour’s channel, http://www.youtube. Page 37 a snub dodecahedron from thirty com/user/theAmatour, instructional videos of balloons . see Mishel Sabbah, “Polyhe- balloon polyhedra since November 6, 2007. See drons: Snub dodecahedron and snub truncated also http://moria.wesleyancollege.edu/faculty/ icosahedron,” Balloon Animal Forum posting, morey/. 291 292 March 7, 2008, http://www.balloon-animals. Page 44 very nicely described in The com/forum/index.php?topic=931.0. See also the Sleepwalkers by Arthur Koestler ... A.Koestler YouTube video http://www.youtube.com/watch? The Watershed [from Koestler’s larger book, The v=Hf4tfoZC L4. Sleepwalkers] (Garden City, NY, Anchor Books, Page 37 Preserves all or most of the sym- 1960). metry of the polyhedron . see Demaine et al., Page 45 In his little book Symmetry, ... above. H. Weyl, Symmetry Princeton, NJ, Princeton Page 37 family of difficult “NP-complete” University Press, 1952. problems . see Demaine et al., above. Page 37 regular polylinks . Alan Holden, Page 45 Felix Klein did a hundred years Orderly Tangles: Cloverleafs, Gordian Knots, ago ... F.Klein, Lectures on the Icosahedron, and Regular Polylinks, Columbia University English trans., 2nd ed., New York, Dover Press, 1983. Publications, 1956. Page 39 the six-pentagon tangle and the Page 51 may be any number except one. four-triangle tangle . see George W. Hart. ... Gr¨unbaum and T. S. Motzkin “The Number “Orderly tangles revisited,” in Mathematical of Hexagons and the Simplicity of Geodesics Wizardry for a Gardner, pages 187–210, 2009, on Certain Polyhedra,” Canadian Journal of and http://www.georgehart.com/orderly-tangles- Mathematics 15, (1963): 744–51. revisited/tangles.htm. Page 37 computer software to find the right thicknesses of the pieces . see Hart, above. Page 39 polypolyhedra ... Robert J. Notes and References for Chapter 4 Lang. “Polypolyhedra in origami,” in Origami3: Proceedings of the 3rd International Meeting of Page 53 two papyri . Gillings, R. J., Origami Science, Math, and Education, pages Mathematics in the Time of the Pharaohs.Cam- 153–168, 2002, and http://www.langorigami. bridge, Mass.: MIT Press, 1972. com/science/polypolyhedra/polypolyhedra.php4. Page 54 volume of the frustrum of a pyramid Page 39 balloon twisting ... Carl D. Gillings, R. J., “The Volume of a Truncated Worth. “Balloon twisting,” http://www.cworth. Pyramid in Ancient Egyptian Papyri,” The Math- org/balloon twisting/, June 5, 2007, and Demaine ematics Teacher 57 (1964):552–55. et al., above. Page 55 singling them out for study ... Page 39 video instructions ... see,inpar- William C. Waterhouse, “The Discovery of the ticular, Morey (above) and Eckhardt (above). Regular Solids,” Archive for History of Exact Sciences 9 (1972):212–21. Page 55 By the time we get to Euclid ... Heath, T. L. The Thirteen Books of Euclid’s Ele- Notes and References for Chapter 3 ments. Cambridge, England: Cambridge Univer- sity Press, 1925; New York: Dover Publications, Page 42 a very nice book on the history of 1956. these things Scientific . B. L. van der Waerden Page 56 “semiregular” solids discovered Awakening, rev. ed (Leiden: Noordhoff Interna- by Archimedes . Pappus. La Collection tional Publishing; New York, Oxford University Math´ematique. P. Ver Eecke, trans. Paris: De Press, 1974). Brouwer/Blanchard, 1933. See also Como, S., Page 42 Federico gives the details of this Pappus of Alexandria and the Mathematics of story. ... P.J.Federico,Descartes on Polyhedra, Late Antiquity, New York: Cambridge U. Press, (New York, Springer-Verlag, 1982). 2000. Notes and References 293 Page 56 They may date from the Page 57 D¨urer made nets for the dodec- Roman period ... Charriere,` “Nouvelles ahedron ... D¨urer, Albrecht, Unterweysung Hypotheses` sur les dodeca´ edres` Gallo-Romains.” der Messung mit dem Zyrkel und Rychtscheyd, Revue Arch´eologique de l’Est et du Centre-Est N¨urnberg: 1525. 16 (1965):143–59. See also Artmann, Benno, Page 58 two “new” “regular” (star) poly- “Antike Darstellungen des Ikosaeders [Antique hedra . Field J. V., “Kepler’s Star Polyhedra,” representations of the icosahedron],” Mitt. Dtsch. Vistas in Astronomy 23 (1979):109–41. Math.-Ver. 13 (2005): 46–50. Page 58 Kepler’s work is constantly cited Page 56 that have been cited by scholars ... Kepler, Johannes, Harmonices Mundi, Conze, Westdeutsche Zeitschrift f¨ur Geschichte Linz, 1619; Opera Omnia, vol. 5, pp. 75–334, und Kunst 11 (1892):204–10; Thevenot, E., “La Frankfort: Heyden und Zimmer, 1864; M. Caspar, mystique des nombres chez les Gallo-Romains, ed., Johannes Kepler Gesammelte Werke,vol.6, dodeca´ edres` boulets et taureauxtricomes,” Re- Munich: Beck, 1938. vue Arch´eolgique de l’Est et du Centre-Est 6 Page 58 the papers of Lebesgue ... (1955):291–95. Lebesgue, H., “Remarques sur les deux premieres` Page 56 they were used as surveying instru- demonstrations du theor´ eme` d’Euler, relatif´ aux ments . Friedrich Kurzweil, “Das Pentagon- polyedres,”` Bulletin de la Soci´et´eMath´ematique dodekaeder des Museum Carnuntinun und seine de France 52 (1924):315–36. Zweckbestimmung,” Carnuntum Jahrbuch 1956 Page 58 the same statement in P´olya ... (1957):23–29. George Polya,´ Mathematical Discovery (New Page 56 the best guess is that they were York: John Wiley and Sons, 1981), combined candle holders . F. H. Thompson, “Dodecahe- ed., vol. 2, p. 154. drons Again,” The Antiquaries Journal 50, pt. I Page 58 nice paper by Peter Hilton (1970):93–96. and Jean Pedersen . P. Hilton and J. Page 56 A large collection of such Pedersen, “Descartes, Euler, Poincare,´ Polya´ polyhedral objects . F. Lindemann, “Zur and Polyhedra,” L’Enseignment Math´ematique Geschichte der Polyeder und der Zahlze- 27 (1981):327–43. ichen,” Sitzungsberichte der Mathematisch- Page 58 an extensive summary of this debate Physikalische Klasse der Koeniglich Akademie ... P.J.Federico,Descartes on Polyhedra. New der Wissenschaften [Munich] 26 (1890):625–783 York: Springer-Verlag, 1982. and 9 plates. Page 58 evident as they were, I had not Page 57 situation during the Renaissance is perceived them . Jacques Hadamard, An Essay extremely complicated . Schreiber, Peter and on the Psychology of Invention in the Mathemat- Gisela Fischer, Maria Luise Sternath, “New light ical Field (Princeton, N.J.: Princeton University on the rediscovery of the Archimedean solids Press, 1945), 51. during the Renaissance.” Archive for History of Page 59 elongated square gyrobicupola Exact Sciences 62 2008:457–467. (Johnson solid J37) . Norman W. Johnson, Page 57 Pacioli actually made polyhedron “Convex Solids with Regular Faces”, Canadian models ... Pacioli, L. Divina Proportione, Journal of Mathematics, 18, 1966, pages Milan: 1509; Milan: Fontes Ambrosioni, 1956. 169–200. Page 57 related to the emergent study of Page 59 we can continue to honor his mem- perspective ... Descarques,P. Perspective, New ory ... Gr¨unbaum, Branko, “An Enduring Error,” York: Van Nostrand Reinhold, 1982. Elemente der Mathematik 64 (2009):89–101. Page 57 Professor Coxeter discusses ... Page 59 Euler, although he found his Coxeter, H. S. M., “Kepler and Mathematics,” formula, was not successful in proving it Chapter 11.3 in Arthur Beer and Peter Beer, eds., . Euler, Leonhard, “Elementa Doctrinae Kepler: FourHundred Years, Vistas in Astronomy, Solidorum,” Novi Commentarii Academiae vol. 18 (1974), pp. 661–70. Scientiarum Petropolitanae 4 (1752–53): 294 109–40 (Opera Mathematica, vol. 26, pp. 71– Page 58 duals of the Archimedean 93); Euler, Leonhard, “Demonstratio Non- polyhedra . Catalan, M. E., “Memoire´ sur nullarum Insignium Proprietatus Quibus Solida la theorie´ des polyedres,”` Journal de l’Ecole´ Imp. Hedris Planis Inclusa Sunt Praedita,” Novi Polytechnique 41 (1865):1–71. Commentarii Academiae Scientiarum Petropoli- Page 60 Max Br¨uckner published ... see tanae 4 (1752–53):140–60 (Opera Mathematica, Br¨uckner, above. vol. 26, pp. 94–108). Page 60 a paper in 1905 of D. M. Y. Page 59 The first proof was provided Sommerville . Sommerville, D. M. Y., “Semi- by Legendre . Legendre, A., El´´ ements de regular Networks of the Plane in Absolute g´eometrie.
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  • A Survey of Folding and Unfolding in Computational Geometry

    A Survey of Folding and Unfolding in Computational Geometry

    Combinatorial and Computational Geometry MSRI Publications Volume 52, 2005 A Survey of Folding and Unfolding in Computational Geometry ERIK D. DEMAINE AND JOSEPH O’ROURKE Abstract. We survey results in a recent branch of computational geome- try: folding and unfolding of linkages, paper, and polyhedra. Contents 1. Introduction 168 2. Linkages 168 2.1. Definitions and fundamental questions 168 2.2. Fundamental questions in 2D 171 2.3. Fundamental questions in 3D 175 2.4. Fundamental questions in 4D and higher dimensions 181 2.5. Protein folding 181 3. Paper 183 3.1. Categorization 184 3.2. Origami design 185 3.3. Origami foldability 189 3.4. Flattening polyhedra 191 4. Polyhedra 193 4.1. Unfolding polyhedra 193 4.2. Folding polygons into convex polyhedra 196 4.3. Folding nets into nonconvex polyhedra 199 4.4. Continuously folding polyhedra 200 5. Conclusion and Higher Dimensions 201 Acknowledgements 202 References 202 Demaine was supported by NSF CAREER award CCF-0347776. O’Rourke was supported by NSF Distinguished Teaching Scholars award DUE-0123154. 167 168 ERIKD.DEMAINEANDJOSEPHO’ROURKE 1. Introduction Folding and unfolding problems have been implicit since Albrecht D¨urer [1525], but have not been studied extensively in the mathematical literature until re- cently. Over the past few years, there has been a surge of interest in these problems in discrete and computational geometry. This paper gives a brief sur- vey of most of the work in this area. Related, shorter surveys are [Connelly and Demaine 2004; Demaine 2001; Demaine and Demaine 2002; O’Rourke 2000]. We are currently preparing a monograph on the topic [Demaine and O’Rourke ≥ 2005].