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Geometry of Flat Origami Triangulations 4 3 4 3 4 3 4 3 6 6 6 6 5 5 5 5 1 2 1 2 1 2 1 2 4 3 4 3 4 3 4 3 6 6 6 6 5 5 5 5 1 2 1 2 1 2 1 2 Bryan Gin-ge Chen & Chris

  1. Geometry of Flat Origami Triangulations 4 3 4 3 4 3 4 3 6 6 6 6 5 5 5 5 1 2 1 2 1 2 1 2 ��������� 4 3 4 3 4 3 4 3 6 6 6 6 5 5 5 5 1 2 1 2 1 2 1 2 Bryan Gin-ge Chen & Chris Santangelo UMass Amherst Physics

  2. Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 J.-H. Na et al., Adv. Mat. 2015 Wood et al, Science 2015

  3. Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeeve http://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/

  4. Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeeve http://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/

  5. Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeeve http://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/

  6. But how much does a crease pattern really tell us? Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeeve http://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/

  7. But how much does a crease pattern really tell us? What does it tell us near the flat state? Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeeve http://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/

  8. Rigid origami as bond-node structure Triangulated V b -gon : V b boundary vertices

  9. Rigid origami as bond-node structure “bird base” Triangulated V b -gon : V b boundary vertices

  10. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 Triangulated V b -gon : V b boundary vertices

  11. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 V int : # of internal vertices Triangulated V b -gon : V b boundary vertices

  12. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 V int : # of internal vertices Triangulated V b -gon : V b boundary vertices 3V int +V b -3: # of folds

  13. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 V int : # of internal vertices Triangulated V b -gon : V b boundary vertices 3V int +V b -3: # of folds

  14. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 V int : # of internal vertices Triangulated V b -gon : V b boundary vertices 3V int +V b -3: # of folds # of degrees of freedom?

  15. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 V int : # of internal vertices Triangulated V b -gon : V b boundary vertices 3V int +V b -3: # of folds # of degrees of freedom? N 0 = 3( V int + V b ) − (3 V int + V b − 3 + V b )

  16. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 V int : # of internal vertices Triangulated V b -gon : V b boundary vertices 3V int +V b -3: # of folds # of degrees of freedom? N 0 = 3( V int + V b ) − (3 V int + V b − 3 + V b ) = V b + 3 =6 + ( V b − 3)

  17. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 V int : # of internal vertices Triangulated V b -gon : V b boundary vertices 3V int +V b -3: # of folds # of degrees of freedom? N 0 = 3( V int + V b ) − (3 V int + V b − 3 + V b ) = V b + 3 =6 + ( V b − 3) rigid body motions

  18. Rigid origami as bond-node structure “bird base” Demaine et al, Graphs and Combinatorics, 2011 V int : # of internal vertices Triangulated V b -gon : V b boundary vertices 3V int +V b -3: # of folds # of degrees of freedom? N 0 = 3( V int + V b ) − (3 V int + V b − 3 + V b ) = V b + 3 =6 + ( V b − 3) generically V b -3 dof rigid body motions

  19. Configuration space near the flat state 4 3 4 3 6 6 5 5 1 2 1 2 BGC and Santangelo, 2017

  20. Configuration space near the flat state 4 3 4 3 6 6 5 5 1 2 1 2 BGC and Santangelo, 2017

  21. Configuration space near the flat state 4 3 4 3 4 branches 6 6 5 5 1 2 1 2 BGC and Santangelo, 2017

  22. BGC and Santangelo, 2017

  23. BGC and Santangelo, 2017

  24. Flat is not generic!

  25. Flat is not generic! V int +1 linear motions!

  26. Flat is not generic! V int +1 linear motions! vs generically 1dof ??

  27. Flat is not generic! V int +1 linear motions! vs generically 1dof ?? Toy example:

  28. Flat is not generic! V int +1 linear motions! vs generically 1dof ?? linear motion Toy example:

  29. Flat is not generic! V int +1 linear motions! vs generically 1dof ?? linear motion Toy example: redundant constraints = ‘self stress’

  30. Flat is not generic! V int +1 linear motions! vs generically 1dof ?? linear motion compression Toy example: tension redundant constraints = ‘self stress’

  31. Self stresses and second-order constraints linear motion compression tension

  32. Self stresses and second-order constraints linear motion compression tension The second-order constraints are in 1 to 1 correspondence with self stresses! Connelly and Whiteley, SIAM J Discrete Math 1996

  33. Self stresses and second-order constraints linear motion compression tension The second-order constraints are in 1 to 1 correspondence with self stresses! u T Ω u = 0 symmetric Ω “stress matrix” Connelly and Whiteley, SIAM J Discrete Math 1996

  34. Self stresses and second-order constraints linear motion u Ω compression tension The second-order constraints are in 1 to 1 correspondence with self stresses! u T Ω u = 0 symmetric Ω “stress matrix” Connelly and Whiteley, SIAM J Discrete Math 1996

  35. Self stresses in flat triangulations BGC and Santangelo, 2017

  36. Self stresses in flat triangulations “wheel stress” BGC and Santangelo, 2017

  37. Self stresses in flat triangulations “wheel stress” BGC and Santangelo, 2017

  38. Self stresses in flat triangulations vertical “wheel stress” u displacements BGC and Santangelo, 2017

  39. Self stresses in flat triangulations vertical “wheel stress” u displacements u T Ω u = 0 symmetric Ω stress matrix BGC and Santangelo, 2017

  40. Self stresses in flat triangulations vertical “wheel stress” u displacements β 4 , 1 β 2 , 3 β 1 , 2 u T Ω u = 0 ψ 1 ψ 3 ψ 2 Gaussian curvature ψ 4 vanishes at each vertex symmetric α 4 , 1 α 2 , 3 Ω α 1 , 2 stress matrix BGC and Santangelo, 2017

  41. Self stresses in flat triangulations vertical “wheel stress” u displacements β 4 , 1 β 2 , 3 β 1 , 2 u T Ω u = 0 ψ 1 ψ 3 ψ 2 Gaussian curvature ψ 4 ⇐ ⇒ vanishes at each vertex symmetric α 4 , 1 α 2 , 3 Ω !! α 1 , 2 stress matrix BGC and Santangelo, 2017

  42. origami n-vertex configuration space u T Ω u = 0 4 3 (n+1)-vector of vertical u 5 displacements 1 2 (n+1)x(n+1) Ω symmetric stress matrix 6 3-dim kernel from isometries BGC and Santangelo, 2017

  43. origami n-vertex configuration space u T Ω u = 0 4 3 (n+1)-vector of vertical u 5 displacements 1 2 (n+1)x(n+1) Ω symmetric stress matrix 6 3-dim kernel from isometries Always exactly one negative eigenvalue! Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997 BGC and Santangelo, 2017

  44. origami n-vertex configuration space u T Ω u = 0 4 3 (n+1)-vector of vertical u 5 displacements 1 2 (n+1)x(n+1) Ω symmetric stress matrix 6 3-dim kernel from isometries Always exactly one negative eigenvalue! Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997 BGC, Theran and Nixon, 2017 BGC and Santangelo, 2017

  45. origami n-vertex configuration space u T Ω u = 0 4 3 (n+1)-vector of vertical u 5 displacements 1 2 (n+1)x(n+1) Ω symmetric stress matrix 6 3-dim kernel from isometries Always exactly one negative eigenvalue! Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997 BGC, Theran and Nixon, 2017 BGC and Santangelo, 2017

  46. What are the two nappes? 4 3 5 1 2 6 BGC and Santangelo, 2017

  47. What are the two nappes? 4 3 5 1 2 6 BGC and Santangelo, 2017

  48. What are the two nappes? 4 3 5 1 2 6 BGC and Santangelo, 2017

  49. What are the two nappes? 4 3 5 1 2 6 BGC and Santangelo, 2017 Demaine et al, Proceedings of the IASS, 2016

  50. What are the two nappes? 4 3 Negative eigenvector 5 maximizes 1 2 Gaussian curvature 6 BGC and Santangelo, 2017 Demaine et al, Proceedings of the IASS, 2016

  51. What are the two nappes? 4 3 5 1 2 6 BGC and Santangelo, 2017 Demaine et al, Proceedings of the IASS, 2016

  52. What are the two nappes? 4 3 5 1 2 6 BGC and Santangelo, 2017 Demaine et al, Proceedings of the IASS, 2016

  53. What are the two nappes? 4 3 5 1 2 6 BGC and Santangelo, 2017 Demaine et al, Proceedings of the IASS, 2016

  54. What are the two nappes? + 4 3 5 1 2 – 6 BGC and Santangelo, 2017 Demaine et al, Proceedings of the IASS, 2016

  55. What are the two nappes? + 4 3 + 5 1 2 – – 6 The two nappes correspond to popped up and popped down configurations! BGC and Santangelo, 2017 Demaine et al, Proceedings of the IASS, 2016 Abel et al, JoCG, 2016; Streinu and Whiteley, 2005

  56. Multiple vertex configuration space 4 3 4 3 6 6 5 5 1 2 1 2 BGC and Santangelo, 2017

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