1
Edge unfolding vs. general unfolding
Image by MIT OpenCourseWare.
See also: http://erikdemaine.org/papers/Ununfoldable/.
2
[Bern, Demaine,
Eppstein, Kuo, Mantler,
Snoeyink 2003]
3
[Bern, Demaine,
Eppstein, Kuo, Mantler,
Snoeyink 2003]
X
X
Image by MIT OpenCourseWare.
4
v0
A
v0
C
B
D
D
A
v3
v2
E
v1
B
x
v4
v1
C
v2
E
v4
v3
Image by MIT OpenCourseWare.
5
v0
v0
D
B
x
A
C
C
D
v2
v1
C
E
A
B
v3
v4
v4
v3
x
v0
E
A
C
B
v1
v2
E
E
D
v3
v2
E
v1
v4
C
Images by MIT OpenCourseWare.
6
v0
v0
A
C
B
D
D
x
A
v3
v2
E
v4
v1
B
v2
v1
E
v2
x0
v1
v3
x1
x4
v4
A
v2
A
v1
B
E
D
v3
x2
v0
x3
A
x1
v4
B
x3
v3
v4
x0
x4
C
C
A
D
A
x2
v0
Images by MIT OpenCourseWare.
7
13
13
36
42
Image by MIT OpenCourseWare.
8
a’
.75
a
d’
a’
d’
2.125
8.4
a
x
d
x
x
d
x
x
x
1.42
45.1
2.56
3
2
b’
b
b
c
45.2
3
c’
b’
2
c
131.8
.75
48.2
c’
x
Image by MIT OpenCourseWare.
9
10
Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, John Iacono,
Stefan Langerman, and Joseph O'Rourke. Used with permission.
[Demaine,
Demaine, Hart,
Iacono,
Langerman,
O’Rourke 2010]
Albrecht Dürer (self-portrait)
11
Melancholia I [1525]
Dürer's self-portrait and Melancholia I are are in the public domain.
“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)
12
Scan of Dürer's The Painter’s Manual are in the public domain.
Image of cuboctahedron
removed due to
copyright restrictions.
[Dürer 1525]
13
Scan of Dürer's The Painter’s Manual are in the public domain.
Image of Snub cube removed due to copyright restrictions.
[Dürer1525]
Scan of Dürer's The Painter’s Manual are in the public domain.
14
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
Image by MIT OpenCourseWare.
[Namiki & Fukuda 1993]
19
D
C
A
B
B
C
C
D
B
A
C
D
B
A
C
D
Image by MIT OpenCourseWare.
20
120
Percent Overlapping Unfoldings
110
100
90
80
70
60
50
40
30
20
10
0
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Number of Vertices
[Schevon 1989]
21
Image by MIT OpenCourseWare.
A
B
Image by MIT OpenCourseWare.
22
A
B
Image by MIT OpenCourseWare.
23
A
B
A
B
A
[O’Rourke 2001]
24
Images by MIT OpenCourseWare.
B
A
A
[O’Rourke 2001]
25
Image by MIT OpenCourseWare.
[O’Rourke 2007]
26
Image by MIT OpenCourseWare.
Image by MIT OpenCourseWare.
[O’Rourke 2007]
27
a’1
a1
a1
b2
b2
b1
B
B
a2
b1
A
a’1
b0
a0
a’0
A’
a2
a0
b0
a’2
Image by MIT OpenCourseWare.
28
A
B
A’
Image by MIT OpenCourseWare.
[Benbernou, Cahn, O’Rourke 2004]
29
Image by MIT OpenCourseWare.
See also http://erikdemaine.org/papers/CCCG98b/.
[Biedl, Demaine, Demaine,
Lubiw, Overmars, O’Rourke,
Robbins,
Whitesides 1998]
30
[Biedl, Demaine, Demaine,
Lubiw, Overmars, O’Rourke,
Robbins,
Whitesides 1998]
31
Image by MIT OpenCourseWare.
See also http://erikdemaine.org/papers/CCCG98b/.
Image by MIT OpenCourseWare.
See also http://erikdemaine.org/papers/Ununfoldable/.
[Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink 2003]
32
x
C
α
α c
b
a
γ
β
Image by MIT OpenCourseWare.
See also http://erikdemaine.org/papers/Ununfoldable/.
A
β
B
33
[Bern, Demaine,
Eppstein, Kuo, Mantler,
Snoeyink 2003]
Image by MIT OpenCourseWare.
x
x
a
b
[Bern, Demaine,
Eppstein, Kuo, Mantler,
Snoeyink 2003]
34
Image by MIT OpenCourseWare.
See also http://erikdemaine.org/papers/Ununfoldable/.
x
C
c
x
a
A
C
B
c
A
b
a
b
B
[Bern, Demaine,
Eppstein, Kuo, Mantler,
Snoeyink 2003]
35
Image by MIT OpenCourseWare.
See also http://erikdemaine.org/papers/Ununfoldable/.
[Bern, Demaine,
Eppstein, Kuo, Mantler,
Snoeyink 2003]
36
Images by MIT OpenCourseWare.
See also http://erikdemaine.org/papers/Ununfoldable/.
MIT OpenCourseWare
http://ocw.mit.edu
6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra
Fall 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.