What other math problems (e.g. NP- complete) are accessible to us? Annie Siya Sun

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Rigid fold origami may pose interesting problems. Most origami is not made with rigid folds(making it a more unusual phenomenon) a rigid fold is one that could be made with sheet metal and hinges- no bending or twisting just a simple hinge fold from flat to not flat. these folds are of interest to industrial designers/manufactures/package designers and are also becoming useful for solarpanels in space (Miura map fold was one used for sending solarpanels into space)/telescope folding etc.

linear algebra could be used in modeling these kinds of folds particularly using isometry rotation (rotations that preserve distance, thus insuring that folds are rigid....because a bend/crease would not preserve distance)

Problems according to this site not much research has been done on the field, particularly when there is more than one vertex involved in the folding- could be of interest. Maybe we could figure out, given a shape, if and how it can be broken down into rigid folds.(most research has been about analysing crease patterns to determine flatfoldability, but it would be interesting to examine a 3d shape instead. Maybe there could be a way to categorize the simple shapes that make up complex shapes that could be modled and tested by ecoli? -http://kahuna.merrimack.edu/~thull/rigid/rigid.html (the book is Oragami Design Secrets by Robert Lang- a leading math guy/origami designer)

interesting topics and vocab:

  • circle river packing-a design technique for uniaxial bases (a base in which all flaps lie along a single axis and all hinges are perpendicular to the axis) that constructs the crease pattern by packing nonoverlapping circles and rivers into a square.
  • box pleating-a style of folding characterized by all folds running at multiples of 45°, with the majority running at multiples of 0° and 90° on a regular grid.
  • oragami bases- basic folds from which many other shapes can be folded
  • crease patterns- really interesting patterns start to show up on paper when you unfold complex oragami shapes

-google book preview: http://books.google.com/books?id=tqGm8LeomyYC&pg=PT4&lpg=PT4&dq=linear+algebra+in+orgigami&source=bl&ots=XpHe8Zp205&sig=fV-izurvanpIUxqC30RnesVfd0E&hl=en&ei=Pd0RSrXmNYqMtgeW0YCICA&sa=X&oi=book_result&ct=result&resnum=5#PPT265,M1

Origami and NP Complete! apparently slight generalizations of map folding problems are np complete (pages 10-11 of this book)"deciding whether a rectangle with horixontal, vertical and diagonal creases can fold by a series of simple folds is NP complete"...need more info on this -from the book origami3 by Thomas Hull -google book preview: http://books.google.com/books?id=_Nw4Hg89qxoC&pg=PA1&lpg=PA1&dq=linear+algebra+in+orgigami&source=bl&ots=0Usk4d8xHc&sig=-Hh8gFCSJHHcDn3C0MN0vmTXjAA&hl=en&ei=Pd0RSrXmNYqMtgeW0YCICA&sa=X&oi=book_result&ct=result&resnum=8#PPA10,M1 this seems like a great collection of papers that relate all aspects of oragami to math

  • This links to a paper showing/proving that flat fold problems can be NP complete. Very interesting

http://reference.kfupm.edu.sa/content/c/o/the_complexity_of_flat_origami__299207.pdf more on np complete and flat folding www.siam.org/pdf/news/579.pdf

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