Hirata 2000
1/2
3-1/2
3
2
1/4
1/2
3
1
2
3-1/2
1
Image by MIT OpenCourseWare.
1
regular octahedron & tetramonohedron
4
5
x
y
8
3
6
7
3
4
y
1
7
2
8
2
x
9
9
0
Image by MIT OpenCourseWare.
2
O’Rourke
regular octahedron & tetramonohedron
Image by MIT OpenCourseWare.
Horiyama & Uehara 2010
3
cube &
tetramonohedron
Horiyama & Uehara 2010
4
Image by MIT OpenCourseWare.
regular icosahedron & 1 : 1.145 : 1.25 tetramonohedron
1
2
5
6
3
6
3
4
7
4
8
7
5
2
9
8
1
10
9
10
Image by MIT OpenCourseWare.
Horiyama & Uehara 2010
5
Unfolding diagram removed due to copyright restrictions.
6
unfoldings of √‾
2 × √‾
2 × 3 √‾
2 box
B
D
EF
A
G
H
B
A C
HJ
I
C
D
B
D
E
EF
J
F
IG
A
G
H
B
A C
HJ
I
B
C
D
D
E
E
J
A
A
F
IG
F
B
C C
D
EF
G
H H
I J
J
I
unfolding of
1 × 2 × 4 box
Timothy Chan, 1999
7
G
5x1x1
3x2x1
Image by MIT OpenCourseWare.
Therese Biedl, 1999
8
8x1x1
5x2x1
Image by MIT OpenCourseWare.
Therese Biedl, 1999
9
[Uehara 2008]
10
Courtesy of Jun Mitani and Ryuhei Uehara. Used with permission.
Courtesy of Jun Mitani and Ryuhei Uehara. Used with permission.
[Uehara 2008]
11
common unfolding of 23 out of 24 pentacubes
Image by MIT OpenCourseWare.
[Aloupis, Benbernou, Bose, Collette, Demaine, Demaine,
Douïeb, Dujmović, Iacono, Langerman, Morin 2010]
12
unique common unfolding of nonplanar pentacubes
(& 22 total pentacubes)
Image by MIT OpenCourseWare.
[Aloupis, Benbernou, Bose, Collette, Demaine, Demaine,
Douïeb, Dujmović, Iacono, Langerman, Morin 2010]
13
common unfolding of planar pentacubes
(& no nonplanar pentacubes)
Image by MIT OpenCourseWare.
[Aloupis, Benbernou, Bose, Collette, Demaine, Demaine,
Douïeb, Dujmović, Iacono, Langerman, Morin 2010]
14
Screenshot of applet showing pentacubes removed due to copyright restrictions.
15
Image by MIT OpenCourseWare.
See also http://arxiv.org/abs/cs/0110059/.
16
[Donoso &
O’Rourke
2002]
Base polyhedron with genus 11 and net, and Polyhedron with genus 6 and
net removed due to copyright restrictions.
Refer to: Fig. 2-4 from Biedl, T., T. M. Chan, et al. “Tighter Bounds on the Genus
of Nonorthogonal Polyhedra Built from Rectangles." Proceedings of the 14th
Canadian Conference on Computational Geometry (2002): 105–8.
17
D-forms
S1
S2
Image by MIT OpenCourseWare.
Tony Wills
18
A
B
A’
Image by MIT OpenCourseWare.
19
[Benbernou, Cahn, O’Rourke 2004]
20
Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.
Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.
21
22
Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.
23
Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.
Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.
24
[Demaine, Demaine, Iacono, Langerman 2009]
MIT OpenCourseWare
http://ocw.mit.edu
6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra
Fall 2012
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