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Class 15 Slides: Polyhedron Unfolding I, 6.849 Fall 2012 Can you formally define what a handle is? 1 Gudenrath Glassblowing: Handle — casting off http://youtu.be/nACHHJwcFWM 2 http://en.wikipedia.org/wiki/Genus_(mathematics) Images of genus examples removed due to copyright restrictions. Image of genus morph of cup is in the public domain. 3 Why do convex polyhedral unfoldings necessarily have no holes? 4 Image by MIT OpenCourseWare. See also: http://erikdemaine.org/papers/Ununfoldable/. [Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003] 5 Can you explain the comment “leaves = the polyhedron vertices” on page 4? It looks like vertices usually have unique shortest paths to ." 풙 6 v0 v0 A D D C B B C x A v v3 2 v2 E v E v4 1 v3 v1 v4 Image by MIT OpenCourseWare. 7 Any luck finding generalizations of star and source unfoldings? 8 Figures of unfoldings removed due to copyright restrictions. Refer to: Fig. 1-4 from Demaine, E. D., and A. Lubiw. "A Generalization of the Source Unfolding of Convex Polyhedra." Revised Papers from the 14th Spanish Meeting on Computational Geometry Lecture Notes in Computer Science 7579 (2011): 185–99. 9 Figures of unfoldings removed due to copyright restrictions. Refer to: Fig. 1-4 from Demaine, E. D., and A. Lubiw. "A Generalization of the Source Unfolding of Convex Polyhedra." Revised Papers from the 14th Spanish Meeting on Computational Geometry Lecture Notes in Computer Science 7579 (2011): 185–99. 10 Figures of unfoldings removed due to copyright restrictions. Refer to: Fig. 1-4 from Demaine, E. D., and A. Lubiw. "A Generalization of the Source Unfolding of Convex Polyhedra." Revised Papers from the 14th Spanish Meeting on Computational Geometry Lecture Notes in Computer Science 7579 (2011): 185–99. 11 Figures of unfoldings removed due to copyright restrictions. Refer to: Fig. 1-4 from Demaine, E. D., and A. Lubiw. "A Generalization of the Source Unfolding of Convex Polyhedra." Revised Papers from the 14th Spanish Meeting on Computational Geometry Lecture Notes in Computer Science 7579 (2011): 185–99. 12 double pyramid octahedron Bridges Exhibition of Mathematical Art Portugal, July 2011 [Demaine, Demaine, Lubiw, Shallit, Shallit 2010–2011] Courtesy of Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit. Used with permission. 13 Courtesy of Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit. Used with permission. [Demaine, Demaine, Lubiw, Shallit, Shallit 2010] 3D images of polyhedra removed due to copyright restrictions. 14 [Demaine, Demaine, Lubiw, Shallit, Shallit 2010] Courtesy of Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit. Used with permission. 3D images of polyhedra removed due to copyright restrictions. 15 [Demaine, Demaine, Lubiw, Shallit, Shallit 2010] Courtesy of Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit. Used with permission. 16 3D images of polyhedra removed due to copyright restrictions. Courtesy of Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit. Used with permission. [Demaine, Demaine, Lubiw, Shallit, Shallit 2010] 3D images of polyhedra removed due to copyright restrictions. 17 [Demaine, Demaine, Lubiw, Shallit, Shallit 2010] 18 Edge unfoldings seem super cool. […] before, they seemed pretty obviously doable but you have convinced me otherwise. 19 Images by MIT OpenCourseWare. See also http://erikdemaine.org/papers/Ununfoldable/. 20 21 Drawings of no-net polyhedron (Fig. 1a) and star-shaped polyhedron (Fig. 2) removed due to copyright restrictions. Refer to: Grünbaum, B. "No-Net Polyhedra." Geombinatorics 11 (2002): 111–14; Grünbaum, B. "A Star-shaped Polyhedron with No Net." Geombinatorics 11 (2001): 43–48. 22 Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. 23 [Abel & Demaine 2011] Courtesy of Zachary Abel and Erik D. Demaine. Used with permission. 24 [Abel & Demaine 2011] Courtesy of Zachary Abel and Erik D. Demaine. Used with permission. 25 [Abel & Demaine 2011] Courtesy of Zachary Abel and Erik D. Demaine. Used with permission. 26 [Abel & Demaine 2011] Courtesy of Zachary Abel and Erik D. Demaine. Used with permission. 27 Images from Pepakura Designer 3 removed due to copyright restrictions. 28 Does a band unfolding work on a prismoid? 29 a’1 a1 a1 b2 b2 b1 B b1 B A a’1 a2 a2 b a a’ 0 0 0 a0 A’ b0 a’2 Image by MIT OpenCourseWare. 30 Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. [Demaine, Demaine, Lubiw 1999] 31 Courtesy of Greg Aloupis. Used with permission. Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. 32 Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. 33 Can you describe in more detail what “continuous blooming” is? Tell us more about continuous blooming. 34 35 Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, John Iacono, Stefan Langerman, and Joseph O'Rourke. Used with permission. 36 Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, John Iacono, Stefan Langerman, and Joseph O'Rourke. Used with permission. 37 MIT OpenCourseWare http://ocw.mit.edu 6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra Fall 2012 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms..
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