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Edge unfolding vs. general unfolding1
[Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003]
Image by MIT OpenCourseWare.See also: http://erikdemaine.org/papers/Ununfoldable/.
2
[Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003]
3
4
XX
Image by MIT OpenCourseWare.
D CC
A
ABB
v2
v2
v4
v4v1
v0v0
v1
v3
v3
EE
Dx
5
Image by MIT OpenCourseWare.
D C
A
B
v2
v4
v1
v0
v3E
x
CA
B
v2 v4
v0
v1
v3
E
D
C
C C
A B
v2
v4
v0
v1
v3
E
E E
D
x
Images by MIT OpenCourseWare.
6
C
B
v2
x2
v4
x4
x0
v0
x1v1
v3
x3
E D
B D
A A
A A
A
x2
v2
v4
x4
x0
v0
v1 x1
v3
x3
CA
B
v2 v4
v0
v1
v3
E
D D C
A
B
v2
v4
v1
v0
v3E
x
Images by MIT OpenCourseWare.7
13 13
36 42
Image by MIT OpenCourseWare.
8
48.2
8.4
45.1
.8
a’ d’
a
b c
b’
d’
a’
a
.75
2.125
c’
x
x
x x
xx
d
1313
c’
c
b’
d
31.42 2.56
2
2
x
b45.2
.75
Image by MIT OpenCourseWare.
9
[Demaine, Demaine, Hart,
Iacono, Langerman,
O’Rourke 2010]Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, John Iacono,Stefan Langerman, and Joseph O'Rourke. Used with permission.10
Albrecht Dürer (self-portrait) Melancholia I [1525]Dürer's self-portrait and Melancholia I are are in the public domain.
11
“The Painter’s Manual: A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler Assembled by Albrecht Dürer for the Use of All Lovers of Art with Appropriate Illustrations Arranged to be Printed in the Year MDXXV (1525)
Scan of Dürer's The Painter’s Manual are in the public domain.12
[Dürer 1525]
Image of cuboctahedron
removed due to
copyright restrictions.
13 Scan of Dürer's The Painter’s Manual are in the public domain.
[Dürer1525]
Image of Snub cube removed due to copyright restrictions.
14Scan of Dürer's The Painter’s Manual are in the public domain.
Images of polyhedra animations and respective unfoldings removed due to copyright restrictions.
15
Image removed due to copyright restrictions.
Refer to: Fig. 40c from Schlickenrieder, W. "Nets of Polyhedra."
Technische Universität Berlin, Masters's Thesis, 1997.
16
Image removed due to copyright restrictions.
Refer to: Fig. 35 from Schlickenrieder, W. "Nets of Polyhedra."
Technische Universität Berlin, Masters's Thesis, 1997.
17
Image removed due to copyright restrictions.
Refer to: Fig. 38, 42b from Schlickenrieder, W. "Nets of Polyhedra."
Technische Universität Berlin, Masters's Thesis, 1997.
18
[Namiki & Fukuda 1993]Image by MIT OpenCourseWare.
19
A
A
B
B
B
B
D
C
C D
D
D
C
C
C
A
Image by MIT OpenCourseWare.
20
[Schevon 1989]
00 5 15
10
10
20
20 25 30 35 40 45 50 55 60 65 70 75 80
30
40
50
60
70
Per
cen
t O
verl
app
ing
Un
fold
ing
s
Number of Vertices
80
90
100
110
120
Image by MIT OpenCourseWare.
21
B
A
Image by MIT OpenCourseWare.22
A
B
Image by MIT OpenCourseWare.
23
[O’Rourke 2001]
A
B
A
A
B
Images by MIT OpenCourseWare.
24
[O’Rourke 2001]
A
A
B
Image by MIT OpenCourseWare.
25
[O’Rourke 2007]Image by MIT OpenCourseWare.
26
[O’Rourke 2007]Image by MIT OpenCourseWare.
27
b2b2 b1b1
a0a0
b0
b0
a1a1
a’1
a’1
a’0
a’2
a2a2
B
B A
A’
Image by MIT OpenCourseWare.28
[Benbernou, Cahn, O’Rourke 2004]
A
B
A’
Image by MIT OpenCourseWare.
29
[Biedl, Demaine, Demaine,Lubiw, Overmars, O’Rourke,Robbins, Whitesides 1998]
Image by MIT OpenCourseWare.See also http://erikdemaine.org/papers/CCCG98b/.
30
[Biedl, Demaine, Demaine,Lubiw, Overmars, O’Rourke,Robbins, Whitesides 1998]
Image by MIT OpenCourseWare.See also http://erikdemaine.org/papers/CCCG98b/.
31
[Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003]
Image by MIT OpenCourseWare.See also http://erikdemaine.org/papers/Ununfoldable/.
32
[Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003]
C
Aa
x
b cαα
β
γ
B
βImage by MIT OpenCourseWare.See also http://erikdemaine.org/papers/Ununfoldable/.
Image by MIT OpenCourseWare.
33
[Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003]
a
x
x
b
Image by MIT OpenCourseWare.See also http://erikdemaine.org/papers/Ununfoldable/.
34
[Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003]
C
Aa
x
bc
BC
Aa
x
bc
BImage by MIT OpenCourseWare.See also http://erikdemaine.org/papers/Ununfoldable/.
35
[Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003]
Images by MIT OpenCourseWare.See also http://erikdemaine.org/papers/Ununfoldable/.
36
MIT OpenCourseWarehttp://ocw.mit.edu
6.849 Geometric Folding Algorithms: Linkages, Origami, PolyhedraFall 2012
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