Flat Folding of Box-Pleated Crease Pattern is NP-complete
[Akitaya, Cheung,
Demaine, Horiyama, Hull,
Ku, Tachi, Uehara 2015]
Not-All-Equal 3SAT
• Given triples• Assign binary
values to variables so that every triple has both values
Not-All-Equal 3SAT
• Given triples• Assign binary
values to variables so that every triple has both values
Not-All-Equal 3SAT
• Given triples• Assign binary
values to variables so that every triple has both values
Not-All-Equal 3SAT
• Given triples• Assign binary
values to variables so that every triple has both values
Not-All-Equal 3SAT
• Given triples• Assign binary
values to variables so that every triple has both values
Graph Orientation
Graph Orientation
[Bern & Hayes 1996]
Presenter
Presentation Notes
Figure 13.3 of GFALOP
[Akitaya, Cheung, Demaine, Horiyama, Hull, Ku, Tachi, Uehara 2015]
Flat Folding of Box-Pleated Crease Pattern is NP-complete
[Akitaya, Cheung,
Demaine, Horiyama, Hull,
Ku, Tachi, Uehara 2015]
Flat Folding of Box-Pleated Crease Pattern is NP-complete
[Akitaya, Cheung,
Demaine, Horiyama, Hull,
Ku, Tachi, Uehara 2015]
[Akitaya, Cheung, Demaine, Horiyama, Hull, Ku, Tachi, Uehara 2015]
Flat Folding of Box-Pleated Assigned Crease Pattern is NP-complete
[Akitaya, Cheung,
Demaine, Horiyama, Hull,
Ku, Tachi, Uehara 2015]
Flat Folding of Box-Pleated Assigned Crease Pattern is NP-complete
[Akitaya, Cheung,
Demaine, Horiyama, Hull,
Ku, Tachi, Uehara 2015]
SODA 1996
[Akitaya, Cheung, Demaine, Horiyama, Hull, Ku, Tachi, Uehara 2015]
true
true
true
true
[Bern & Hayes 1996]
≠
≠
Crossover is Broken
Electrabrake Manual FolderPress & Shear
Presenter
Presentation Notes
Frames from video http://www.youtube.com/watch?v=l_stw54dfZ4
Luxury and Elegant Home Design In The World / designinteriorart.comJuly 2011
Presenter
Presentation Notes
http://designinteriorart.com/wp-content/uploads/2011/07/unique-wall-decorating-with-various-functions.jpg and http://designinteriorart.com/wp-content/uploads/2011/07/wall-decorating-ideas-with-various-functions.jpg from http://designinteriorart.com/all-posts/decorating-wall-with-various-functions/
“Sheetseat”Ufuk Keskin & Efecem Kutuk2009
Presenter
Presentation Notes
http://www.ufukkeskin.com/storage/sheetseat4.jpg from http://www.ufukkeskin.com/sheetseat/
“Flat-Pack Furniture” dornob
Presenter
Presentation Notes
http://cdn.dornob.com/wp-content/uploads/2009/04/transforming-all-in-one-room-design3.jpg & http://cdn.dornob.com/wp-content/uploads/2009/04/transforming-fold-out-room-design2.jpg & http://cdn.dornob.com/wp-content/uploads/2009/04/transforming-modular-cardboard-room1.jpg from http://dornob.com/flat-pack-furniture-for-an-all-in-one-interior-design/
http://homegallerydesign.com/wp-content/uploads/Modern-Furniture-Ideas-Unique-Coffee-Table.jpg & http://homegallerydesign.com/wp-content/uploads/Unique-Coffee-Table-Design-Furniture-Ideas.jpg & http://homegallerydesign.com/wp-content/uploads/Unique-Coffee-Table-Design-Folded-Steel.jpg & http://homegallerydesign.com/wp-content/uploads/Unique-Coffee-Table-Design-Home-Furniture.jpg from http://homegallerydesign.com/folded-steel-design-furniture-ideas-unique-coffee-table/
SimpleFolds
[Akitaya, Demaine, Ku 2015]
ModelAssigned Unassigned
One-layer NP-hard open poly open open poly
Some-layers NP-hard
NP-hard poly open NP-
hard poly
All-layers NP-hard
NP-hard poly open NP-
hard poly
InfiniteOne-layer
NP-hard open poly NP-
hard open poly
InfiniteSome-layers
NP-hard open poly NP-
hard open poly
InfiniteAll-layers open open poly open open poly
[Akitaya, Demaine, Ku 2015]
[Arkin, Bender,
Demaine, Demaine, Mitchell,
Sethia, Skiena2000]
Presenter
Presentation Notes
Figure 14.5 of GFALOP
[Arkin, Bender,
Demaine, Demaine, Mitchell,
Sethia, Skiena2000]
Presenter
Presentation Notes
Figures 14.5 and 14.6 of GFALOP
3-Partition
• Given 𝑛𝑛 integers 𝑎𝑎1, 𝑎𝑎2, … , 𝑎𝑎𝑛𝑛, can you partition into 𝑛𝑛/3 triples with the same sum? 𝑡𝑡 = ∑𝑖𝑖 𝑎𝑎𝑖𝑖
⁄𝑛𝑛 3
This problem is strongly NP-complete:NP-complete even if 𝑎𝑎𝑖𝑖 numbers are 𝑛𝑛𝑂𝑂(1)
…?
𝑡𝑡
𝑛𝑛/3 triples
[Akitaya,Demaine,Ku 2015]
Reduction from3-Partition
Reduction from3-Partition
[Akitaya, Demaine, Ku 2015]
[Akitaya, Demaine, Ku 2015]
[Akitaya,Demaine,Ku 2015]
[Demaine, Demaine, Mitchell 1999]3D model byBitmapworld
Presenter
Presentation Notes
3D model from http://www.turbosquid.com/3d-models/3d-horse-animals-lightwave-model/269712 and rendered by Erik. Other drawings by Erik.
Presenter
Presentation Notes
Figure 15.4 of GFALOP
Presenter
Presentation Notes
Figure 15.2 from GFALOP
Presenter
Presentation Notes
Figures 15.2 and 15.3 from GFALOP
Tomohiro Tachi
Origamizer[Tachi 2006;
Demaine & Tachi2009–2016]
Presenter
Presentation Notes
Photos from http://graphics.stanford.edu/data/3Dscanrep/bunny-scanalyze-sh.jpg and http://www.flickr.com/photos/tactom/1003649096/ and http://www.flickr.com/photos/tactom/1002688931
Demaine & Tachi 2010
Presenter
Presentation Notes
From unpublished Origamizer paper by Erik Demaine & Tomohiro Tachi
http://www.flickr.com/photos/tactom/
“Origamizer Screenshots for Hypar”Tomohiro Tachi, 2007
Presenter
Presentation Notes
Top image http://farm2.staticflickr.com/1208/1420669885_e335a62591_o.jpg from http://www.flickr.com/photos/tactom/1420669885/ distributed under Creative Commons Attribution-NonCommercial 2.0 License. Bottom image http://farm2.staticflickr.com/1018/1421577530_49be0f2b12_o.png from http://www.flickr.com/photos/tactom/1421577530/ distributed under Creative Commons Attribution-NonCommercial 2.0 License.
“3D origamibell shape”
Tomohiro Tachi2007
http://www.flickr.com/photos/tactom/
Presenter
Presentation Notes
Image http://farm2.staticflickr.com/1144/913624865_c0c90c4dbf_o.jpg from http://www.flickr.com/photos/tactom/913624865/ distributed under Creative Commons Attribution-NonCommercial 2.0 License
“Mouse”Tomohiro Tachi
2007
http://www.flickr.com/photos/tactom/
Presenter
Presentation Notes
Image http://farm3.staticflickr.com/2399/2330200221_6b0f364462_o.jpg from http://www.flickr.com/photos/tactom/1476518003/ distributed under Creative Commons Attribution-NonCommercial 2.0 License
“3D mask”Tomohiro Tachi
2007
http://www.flickr.com/photos/tactom/
Presenter
Presentation Notes
Image http://farm2.staticflickr.com/1251/1476518003_1a816a5e0d_o.jpg from http://www.flickr.com/photos/tactom/2330200221/ distributed under Creative Commons Attribution-NonCommercial 2.0 License
“Tetrapod”Tomohiro Tachi
2008
http://www.flickr.com/photos/tactom/
Presenter
Presentation Notes
Left image http://farm4.staticflickr.com/3003/2653269033_6fb887c21c_o.jpg from http://www.flickr.com/photos/tactom/2653269033/ distributed under Creative Commons Attribution-NonCommercial 2.0 License. Right image http://freeassociationdesign.files.wordpress.com/2010/09/2802993534_2de3ba2d37_b.jpg from http://freeassociationdesign.wordpress.com/2010/09/18/tetrapods-entropy-and-excess/
“Leaf of Kajinoki(BroussonetiaPapyrifera)”
Tomohiro Tachi2007
http://www.flickr.com/photos/tactom/
Presenter
Presentation Notes
Image http://farm2.staticflickr.com/1261/1426795877_e4a33a5617_o.jpg from http://www.flickr.com/photos/tactom/1426795877/ distributed under Creative Commons Attribution-NonCommercial 2.0 License
Presenter
Presentation Notes
Video: http://www.youtube.com/watch?v=GAnW-KU2yn4
“OrigamiStanfordBunny”
Tomohiro Tachi2007
http://www.flickr.com/photos/tactom/
Presenter
Presentation Notes
Image http://farm2.staticflickr.com/1284/1002688931_e2b6af10ab_b.jpg from http://www.flickr.com/photos/tactom/1002688931/ distributed under Creative Commons Attribution-NonCommercial 2.0 License
Metal Bunny
[Cheung, Demaine, Demaine, Tachi 2011]
Presenter
Presentation Notes
Video http://www.flickr.com/photos/tactom/5473218888/ distributed under Creative Commons Attribution-NonCommercial 2.0 License
Metal Bunny
[Cheung, Demaine, Demaine, Tachi 2011]
Presenter
Presentation Notes
Photo http://farm6.staticflickr.com/5292/5473218708_44294474ec_b.jpg from http://www.flickr.com/photos/tactom/5473218708/ distributed under Creative Commons Attribution-NonCommercial 2.0 License
Tree Method of Origami Design[Fujimoto, Kamiya, Kawahata, Lang, Maekawa, Meguro, Yoshino]
[Lang, Demaine, Demaine 2006–2016]
• Algorithm to fold optimal origami “base” with desired stick-figure projection
headforeleg
hindleg
tail
Tree Method of Origami Design[Fujimoto, Kamiya, Kawahata, Lang, Maekawa, Meguro, Yoshino]
[Lang, Demaine, Demaine 2006–2016]
• Algorithm to fold optimal origami “base” with desired stick-figure projection
headforeleg
hindleg
tail
“Scorpion varileg, opus 379”Robert Lang, 2002
Presenter
Presentation Notes
Top colored figure is Figure 11.35 from Origami Design Secrets (1st edition), page 406. Tree is drawn by Erik. Bottom image http://www.langorigami.com/image.php?image=/art/arachnids/scorpion_varileg.jpg from http://www.langorigami.com/art/gallery/gallery.php?name=scorpion_varileg
“Flying Grasshopper, opus 382”Robert Lang, 2003
Presenter
Presentation Notes
Top colored figure is Figure 11.36 from Origami Design Secrets (1st edition), page 407. Tree is drawn by Erik. Bottom image http://www.langorigami.com/image.php?image=/art/insects/flying_grasshopper.jpg from http://www.langorigami.com/art/gallery/gallery.php?name=flying_grasshopper
“Alamo Stallion, opus 384”Robert Lang, 2002
Presenter
Presentation Notes
Top colored figure is Figure 11.37 from Origami Design Secrets (1st edition), page 408. Tree is drawn by Erik. Bottom image http://www.langorigami.com/image.php?image=/art/mammals/alamo_stallion.jpg from http://www.langorigami.com/art/gallery/gallery.php?name=alamo_stallion
Nazgul 7.3Jason Ku, 2007
Presenter
Presentation Notes
Images and design by Jason Ku. http://scripts.mit.edu/~jasonku/index.php?id=2007&model=nazgul2
Shrimp 1.8Jason Ku, 2006
Presenter
Presentation Notes
Images and design by Jason Ku. http://scripts.mit.edu/~jasonku/index.php?id=2006&model=shrimp1
Butterfly 2.2Jason Ku, 2008
Presenter
Presentation Notes
Images and design by Jason Ku. http://scripts.mit.edu/~jasonku/index.php?id=2008&model=butterfly1
Disk Packing for Origami Design
Disk Packing is NP-hard +𝑎𝑎𝑖𝑖
+𝑎𝑎𝑗𝑗
+𝑎𝑎𝑘𝑘
𝑎𝑎𝑖𝑖 + 𝑎𝑎𝑗𝑗+𝑎𝑎𝑘𝑘 = 𝑡𝑡
[Demaine, Fekete, Lang
2010]−𝑡𝑡
Presenter
Presentation Notes
Figures 10, 11, 2 of http://erikdemaine.org/papers/DiskPacking_Origami5/ (covered under MIT Faculty Open-Access Policy)
Wakoku Chiyekurabe — Kan Chu Sen, 1721
Presenter
Presentation Notes
“Wakoku Chiyekurabe” by Kan Chu Sen, published in 1721. Out of copyright. Scans by Erik Demaine. See http://erikdemaine.org/foldcut/sen_book.html
Wakoku Chiyekurabe — Kan Chu Sen, 1721
Presenter
Presentation Notes
“Wakoku Chiyekurabe” by Kan Chu Sen, published in 1721. Out of copyright. Scans by Erik Demaine. See http://erikdemaine.org/foldcut/sen_book.html
Kan Chu Sen (1721)
Presenter
Presentation Notes
Earliest known reference Precedes earliest origami reference!
“Betsy Ross 1777”Jean Leon Gerome
Ferris, 1932
Harper’s New Monthly Magazine, 1873
1922
Presenter
Presentation Notes
Where Martin Gardner heard of the problem
1955
Presenter
Presentation Notes
Where Martin Gardner heard of the problem
Presenter
Presentation Notes
Figure 17.2 of GFALOP. See also http://erikdemaine.org/foldcut/examples/
Presenter
Presentation Notes
Figure 17.2 of GFALOP. See also http://erikdemaine.org/foldcut/examples/
Presenter
Presentation Notes
Figure 17.2 of GFALOP. See also http://erikdemaine.org/foldcut/examples/
Presenter
Presentation Notes
Figure 17.2 of GFALOP. See also http://erikdemaine.org/foldcut/examples/
Presenter
Presentation Notes
Drawing by Erik Demaine & Martin Demaine. http://erikdemaine.org/foldcut/examples/
Jie Qi2010
Sarah Eisenstat2010
Jason Ku2010
Presenter
Presentation Notes
Figure 17.15 of GFALOP
Presenter
Presentation Notes
Figure 1 of http://erikdemaine.org/papers/SimpleFoldCut_CGGA2010full/
Presenter
Presentation Notes
Figure 4 of http://erikdemaine.org/papers/MazeFolding_Origami5/ (covered under MIT Faculty Open-Access Policy)
[Demaine, Demaine, Ku 2010]
Presenter
Presentation Notes
Figure 2 of http://erikdemaine.org/papers/MazeFolding_Origami5/ (covered under MIT Faculty Open-Access Policy)
[Demaine, Demaine, Ku 2010]
Presenter
Presentation Notes
Made with http://erikdemaine.org/fonts/maze/
[Demaine, Demaine, Ku 2010]
Presenter
Presentation Notes
Figure 4 of http://erikdemaine.org/papers/MazeFolding_Origami5/ (covered under MIT Faculty Open-Access Policy)
folding by Christopher Chin
Presenter
Presentation Notes
Figure 4 of http://erikdemaine.org/papers/MazeFolding_Origami5/ (covered under MIT Faculty Open-Access Policy)
http://erikdemaine.org/fonts/
Presenter
Presentation Notes
http://erikdemaine.org/fonts/maze/
Presenter
Presentation Notes
Class poster from http://courses.csail.mit.edu/6.849/fall12/poster_letter_lowres.pdf . Make your own make folding here: http://erikdemaine.org/fonts/maze/
Yes/No
Demaine,Demaine,
Stengle2011
Presenter
Presentation Notes
http://erikdemaine.org/prints/YesNo/
Science/Art Demaine & Demaine, 2011
Presenter
Presentation Notes
http://erikdemaine.org/prints/ScienceArt/
MartinGardner
Demaine,Demaine
2012
Presenter
Presentation Notes
http://erikdemaine.org/prints/MartinGardner/
“Ralf Konrad'sRubik's Cube Tessellation”
Jorge Jaramillo/ georigami
January 2007
Presenter
Presentation Notes
Photo from http://www.flickr.com/photos/georigami/374615107/ distributed under Creative Commons Attribution 2.0 license