Is there still no software for the fold-and-cut problem? I was totally expecting you to pull out some cool app for it.

Transcription

1 Is there still no software for the fold-and-cut problem? I was totally expecting you to pull out some cool app for it. 1

2 Crease pattern for "The big fish: step by step" removed due to copyright restrictions. Courtesy of David Benjamin and Anthony Lee. Used with permission. David Benjamin & Anthony Lee

3 For the universality condition, don t you need some condition like that every vertex not on the edge of the paper has even degree? Or can the cut be a line segment instead of a whole line? 3

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5 Could you quickly show the process of turning linear corridors into a tree? I am confused about the correspondence between trees and corridors and their relation to being flat foldable. 5

6 B C D E D F D D C G H E F A C B D G G J GH I H G H J I scale: 1X Image by MIT OpenCourseWare. 6

7 On the probability of bad straight skeletons, I was wondering it if should be the other way around [ ] most randomly generated foldcut problems will exhibit this chaotic straight skeleton, since the number of irrational numbers in any interval is more than the number of rational numbers? 7

8 Image by MIT OpenCourseWare. 8

9 Image by MIT OpenCourseWare. 9

10 Image by MIT OpenCourseWare. 10

11 Courtesy of Jie Qi. Used with permission. Jie Qi

12 Crease pattern of witch hat removed due to copyright restrictions. Sarah Eisenstat

13 Courtesy of Jason Ku. Used with permission. Jason Ku

14 Diagram of fold instruction and final figure removed due to copyright restrictions. Refer to: Fig. 6 from Demaine, Erik D., and Martin L. Demaine. "Fold-and-Cut Magic." Tribute to a Mathemagician. A K Peters, 2004, pp

15 Diagram of fold instruction and final figure removed due to copyright restrictions. Refer to: Fig. 6 from Demaine, Erik D., and Martin L. Demaine. "Fold-and-Cut Magic." Tribute to a Mathemagician. A K Peters, 2004, pp

16 Diagram of fold instruction and final figure removed due to copyright restrictions. Refer to: Fig. 4 from Demaine, Erik D., and Martin L. Demaine. "Fold-and-Cut Magic." Tribute to a Mathemagician. A K Peters, 2004, pp

17 I got a little confused in the disk-packing method, and specifically how you went from a disk-packing to the diagram with all the triangles. All in all, cool proof though. 17

18 Image by MIT OpenCourseWare. 18

19 There seem to be many different ways to allocate disks that satisfy the given conditions. Is any allocation good enough to use for a crease pattern? Is there some kind of optimal way of disk packing? 19

20 How related is the tree method to the methods for this is it just that rabbit ears and such come up everywhere, or do we actually have some sort of isomorphism (or not-quiteisomorphism) between foldand-one-cut designs and uniaxial bases? 20

21 Image removed due to copyright restrictions. Refer to: Fig from Lang, Robert J. Origami Design Secrets: MathematicalMethods for an Ancient Art. 1st ed. A K Peters / CRC Press, Image by MIT OpenCourseWare. fold & cut (disk packing method) universal molecule (tree method) 21

22 Image removed due to copyright restrictions. Refer to: Fig from Lang, Robert J. Origami Design Secrets: MathematicalMethods for an Ancient Art. 1st ed. A K Peters / CRC Press, Image by MIT OpenCourseWare. See also Fold-and-Cut Examples: fold & cut (straight skeleton method) universal molecule (tree method) 22

23 Cover and index image removed due to copyright restrictions. Refer to: Lang, Robert J. Origami Design Secrets: Mathematical Methods for an Ancient Art. 2nd ed. A K Peters / CRC Press,

24 Figure removed due to copyright restrictions. Refer to: Fig from Lang, Robert J. Origami Design Secrets: Mathematical Methods for an Ancient Art. 2nd ed. A K Peters / CRC Press,

25 I wonder how much this changes if you change from having a straight cut to one curved cut of fixed radius. Presumably, all your edges have to have that curve, but can you get interesting shapes out of it? 25

26 The explanation you gave a couple classes ago about what higherdimensional folding means made sense to me at the time, but it makes considerably less sense now that you mentioned (this lecture) that you need to fold through the fourth dimension when you fold a grocery bag flat. [ ] How do you explain this disconnect between theory and reality? 26

27 Image by MIT OpenCourseWare. 27

28 [Itoh, Nara, Vîlcu 2011] v 8 v 5 = v 1 h 4 v 8 = v 4 g h 3 v 7 = v 3 v 5 h 1 g 4 h 4 v 4 v 6 g 3 g 2 v 7 h 3 h1 v 6 = v 2 h 2 v 1 g 1 h 2 v 3 v 2 v 17 v 16 v 17 v 16 v 17 v v v 15 v 18 v 20 v v11 g 3 18 v v 20 v v v15 10 v 18 v 15 v 5i+4 v 12 v 4 g 4 g 2 v v12 v v v 11 v v g 1 14 g 5 v 9 v 6 v 9 v 6 v 4 v v 6 v 9 v v v 4 13 v 13 v 19 v 5i+5 g v 5i+3 v 5 v v 3 v 7 v 8 5 v 8 v 3 v 5 v v v v v 5i+1 v 5i+2 v 1 v 2 v 1 v 2 v 1 v 2 Images by MIT OpenCourseWare. 28

29 Courtesy of Erik Demaine and Martine Demaine. Used with permission. [Demaine & Demaine 2002] 29

30 Impenetrable Castle Peter Callesen 2005 Courtesy of Peter Callesen. Used with permission. 30

31 Distant Wish Peter Callesen 2006 Courtesy of Peter Callesen. Used with permission. 31

32 The Dying Swan II Peter Callesen 2010 Courtesy of Peter Callesen. Used with permission. 32

33 The End of the Road Peter Callesen 2010 Courtesy of Peter Callesen. Used with permission. 33

34 Running Fire II Peter Callesen 2010 Courtesy of Peter Callesen. Used with permission. 34

35 Closet Peter Callesen 2006 Courtesy of Peter Callesen. Used with permission. 35

36 Pandora s Box Peter Callesen 2005 Courtesy of Peter Callesen. Used with permission. 36

37 White Hand Peter Callesen 2007 Courtesy of Peter Callesen. Used with permission. 37

38 The Short Distance Between Time and Shadow Peter Callesen 2006 Courtesy of Peter Callesen. Used with permission. 38

39 Down the River Peter Callesen 2005 Courtesy of Peter Callesen. Used with permission. 39

40 Angel Peter Callesen 2006 Courtesy of Peter Callesen. Used with permission. 40

41 MIT OpenCourseWare Geometric Folding Algorithms: Linkages, Origami, Polyhedra Fall 2012 For information about citing these materials or our Terms of Use, visit: