slides for class 14 (pdf - 2.2mb)
TRANSCRIPT
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Is there software for hinged dissections?
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[Mabry 2005]
Courtesy of Rick Mabry. Used with permission.
Refer to: http://lsusmath.rickmabry.org/rmabry/live3d/hinged-triangle-pentagon.htm.2
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Cover of books removed due to copyright restrictions.
Refer to: Frederickson, Greg N. Dissections: Plane & Fancy. Cambridge University Press, 1997. Frederickson, Greg N. Hinged Dissections: Swinging & Twisting. Cambridge University Press, 2002. Frederickson, Greg N. Piano-Hinged Dissections. Cambridge University Press, 2006.
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Alphabet Hinged Dissection [Demaine & Demaine 2003]
• 32-abolos, 128 pieces
“Hinged alphabet” Erik & Martin Demaine
2003
Courtesy of Erik Demaine and Martin Demaine. Used with permission.4
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[Demaine, Demaine, Eppstein, Frederickson, Friedman 1999/2005] 5
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[Demaine, Demaine, Eppstein, Frederickson, Friedman 1999/2005]
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.6
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[Demaine, Demaine, Eppstein, Frederickson, Friedman 1999/2005]
1
2
3
25
2627
28
22
9
10
11
1213
21
74 5
24 23
16
15
68
20141918
17
24
2325
27 2628
1 4
32 6 197
8 18
1716
22
21520
10
9
11 15
141213
Image by MIT OpenCourseWare.
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[Demaine, Demaine, Eppstein,
Frederickson, Friedman
1999/2005]
Image by MIT OpenCourseWare.8
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[Demaine, Demaine, Lindy, Souvaine 2005]
Octaherdron
Icosahedron Cube Tetrahedron
Dodecahedron
Image by MIT OpenCourseWare.
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1cm
[Mao, Thalladi, Wolfe, Whitesides, Whitesides 2002]
Images removed due to copyright restrictions. Refer to: Fig. 2, 3 from Chengede, M., V. R. Thalladi. "Communication Dissections:
Self-Assembled Aggregates That Spontaneously Reconfigure Their Structures When
Their Environment Changes." J. Am. Chem. Soc. 124, no. 49 (2002): 14508–9.
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~100nm
[Rothemund — Nature 2006] Courtesy of Paul W. K. Rothemund. Used with permission.
Courtesy of Nature Publishing Group. Used with permission.
Courtesy of Paul W. K. Rothemund. Used with permission.11
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Images of turning conditions, Minkowski sums, and Hamiltonian paths removed due to copyright restrictions.
Refer to: Fig. 3, 4 from Cheung, K. C., E. D. Demaine, et al. "Programmable Assembly With Universally
Foldable Strings (Moteins)." IEEE Transactions on Robotics 27, no. 4 (2011): 718–29.
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Macrobot MIT Center for Bits & Atoms Courtesy of Skylar Tibbits. Used with permission.To view video: http://vimeo.com/4855377.
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Decibot, 144′′ × 18′′ × 18′′ MIT Center for Bits & Atoms
Courtesy of Skylar Tibbits. Used with permission.
To view video: http://vimeo.com/8002813.14
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Crystalline (3x) [Dartmouth/ MIT 2001]
Molecubes (4x) [Cornell 2005] M-TRAN (8x) [Tokyo 2002–2006]
Photograph of robots removed due to copyright restrictions. Refer to: CSAIL wiki for the Self-Reconfiguring Crystal Robot;
webpage for M-TRAN; video of Molecubes.
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Can we see the dissection of a rectangle into a rectangle of another height?
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[Montucla] [Ozanam 1778]
[Frederickson 1997]
R’
R
Image by MIT OpenCourseWare.
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[Dudeney 1902] [Frederickson 2008]
Frederickson, Greg N. "Designing a Table Both Swinging and Stable." The College Mathematics Journal
39, no. 4 (2008): 258-66. Copyright (c) 2008 Mathematical Association of America. Used with permission.
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[Frederickson 2008] Frederickson, Greg N. "Designing a Table Both Swinging and Stable." The College Mathematics Journal
39, no. 4 (2008): 258-66. Copyright (c) 2008 Mathematical Association of America. Used with permission.
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[Frederickson 2008]
Frederickson, Greg N. "Designing a Table Both Swinging and Stable." The College Mathematics Journal
39, no. 4 (2008): 258-66. Copyright (c) 2008 Mathematical Association of America. Used with permission.
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[Frederickson 2008]
Frederickson, Greg N. "Designing a Table Both Swinging and Stable." The College Mathematics Journal
39, no. 4 (2008): 258-66. Copyright (c) 2008 Mathematical Association of America. Used with permission.
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[Frederickson 2008]
Frederickson, Greg N. "Designing a Table Both Swinging and Stable." The College Mathematics Journal
39, no. 4 (2008): 258-66. Copyright (c) 2008 Mathematical Association of America. Used with permission.
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[Frederickson 2008]
Still from animation of hinging triangle to square removed due to copyright restrictions. To view animation: http://www.cs.purdue.edu/homes/gnf/book2/anim34SS.mpg.
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[D*Haus Company Ltd, 2012]
Photograph removed due to copyright restrictions.
To view video: http://vimeo.com/37822655.
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[D*Haus Company Ltd, 2011]
Photograph removed due to copyright restrictions.
To view video: http://vimeo.com/30108578.
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For step 3 of hinged dissections, you said that the number of pieces roughly doubles at each step, but from the diagrams it looks like the number of pieces would more than double.
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[Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2010]
Courtesy of Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine,
Martin L. Demaine, and Scott Duke Kominers. Used with permission.
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[Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2010]
Courtesy of Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine,
Martin L. Demaine, and Scott Duke Kominers. Used with permission. 28
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I’m curious about the pseudopolynomial bound for hinged dissection into a long rectangle, like you briefly mentioned.
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[Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2010]
Courtesy of Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine,
Martin L. Demaine, and Scott Duke Kominers. Used with permission.
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[Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2010]
Courtesy of Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine,
Martin L. Demaine, and Scott Duke Kominers. Used with permission.
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[Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2010]
Courtesy of Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine,
Martin L. Demaine, and Scott Duke Kominers. Used with permission.
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[Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2010]
Courtesy of Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine,
Martin L. Demaine, and Scott Duke Kominers. Used with permission.
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[Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2010]
Courtesy of Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine,
Martin L. Demaine, and Scott Duke Kominers. Used with permission.
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Can we get a brief overview of 3D dissections?
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[Hilbert 1900] Excerpted from Hilbert, David. “Mathematical Problems.” Bulletin of the American Mathematical Society 8,no. 10 (1902): 437-79. Courtesy of the American Mathematical Society. Used with permission.36
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