Download presentation

1
**Part II: Paper a: Flat Origami**

Joseph O’Rourke Smith College (Many slides made by Erik Demaine)

2
**Outline Single-vertex foldability Continuous Foldability**

Kawaskai’s Theorem Maekawa’s Theorem Justin’s Theorem Continuous Foldability Map Folding (revisited)

3
**Foldings Piece of paper = 2D surface**

Square, or polygon, or polyhedral surface Folded state = isometric “embedding” Isometric = preserve intrinsic distances (measured along paper surface) “Embedding” = no self- intersections except that multiple surfaces can “touch” with infinitesimal separation Nonflat folding Flat origami crane

4
Foldings Configuration space of piece of paper = uncountable-dim. space of all folded states Folding motion = path in this space = continuum of folded states Fortunately, configuration space of a polygonal paper is path-connected [Demaine, Devadoss, Mitchell, O’Rourke 2004] Focus on finding interesting folded state

5
Structure of Foldings Creases in folded state = discontinuities in the derivative Crease pattern = planar graph drawn with straight edges (creases) on the paper, corresponding to unfolded creases Mountain-valley assignment = specify crease directions as or Nonflat folding Flat origami crane

6
**Flat Foldings of Single Sheets of Paper**

7
**Single-Vertex Origami**

Consider a disk surrounding a lone vertex in a crease pattern (local foldability) When can it be folded flat? Depends on Circular sequence of angles between creases: Θ1 + Θ2 + … + Θn = 360° Mountain-valley assignment

8
**Single-Vertex Origami without Mountain-Valley Assignment**

Kawasaki’s Theorem: Without a mountain-valley assignment, a vertex is flat-foldable precisely if sum of alternate angles is 180° (Θ1 + Θ3 + … + Θn−1 = Θ2 + Θ4 + … + Θn) Tracing disk’s boundary along folded arc moves Θ1 − Θ2 + Θ3 − Θ4 + … + Θn−1 − Θn Should return to starting point equals 0

9
**Single-Vertex Origami with Mountain-Valley Assignment**

Maekawa’s Theorem: For a vertex to be flat-foldable, need |# mountains − # valleys| = 2 Total turn angle = ±360° = 180° × # mountains − 180° × # valleys

10
**Single-Vertex Origami with Mountain-Valley Assignment**

Kawasaki-Justin Theorem: If one angle is smaller than its two neighbors, the two surrounding creases must have opposite direction Otherwise, the two large angles would collide These theorems essentially characterize all flat foldings

11
**Local Flat Foldability**

Locally flat-foldable crease pattern = each vertex is flat-foldable if cut out = flat-foldable except possibly for nonlocal self-intersection Testable in linear time [Bern & Hayes 1996] Check Kawasaki’s Theorem Solve a kind of matching problem to find a valid mountain-valley assignment, if one exists Barrier:

12
**Global Flat Foldability**

Testing (global) flat foldability is strongly NP-hard [Bern & Hayes 1996] Wire represented by “crimp” direction: False True T T F T F T T T F T F T self-intersects Not-all-equal 3-SAT clause

13
Continuous Rolling

14
**Continuous Foldability**

15
Models of Simple Folds A single line can admit several different simple folds, depending on # layers folded Extremes: one-layer or all-layers simple fold In general: some-layers simple fold Example in 1D:

16
**Simple Foldability [Arkin, Bender, Demaine, Demaine, Mitchell, Sethia, Skiena 2001]**

= = =

17
Open Problems Open: Orthogonal creases on non-axis-aligned rectangular piece of paper? Open (Edmonds): Complexity of deciding whether an m × n grid can be folded flat (has a flat folded state) with specified mountain-valley assignment Even 2xN maps open! Open: What about orthogonal polygons with orthogonal creases, etc.? NP-Complete?

Similar presentations

OK

Foldable 1. Take out a piece of notebook paper and make a hot dog fold over from the right side over to the pink line.

Foldable 1. Take out a piece of notebook paper and make a hot dog fold over from the right side over to the pink line.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on indian army weapons pictures Ppt on adding and subtracting matrices Ppt on world ending in 2012 Ppt on swami vivekananda speech Ppt on 2 dimensional figures and 3 dimensional slides shoes A ppt on loch ness monster photos Ppt on fundamental rights of india Ppt on pin diode limiters Ppt on microsoft word 2003 Ppt on contract labour act 1970 malaysia