Hirata 200

3

2

2

1

1

1/41/2

1/2

3-1/2

3

3-1/2

0

Image by MIT OpenCourseWare.

1

regular octahedron & tetramonohedron

O’Rourke

2 9

7

x

y

8

4

3

12

67

x

y

8

9 0

5 4

3

Image by MIT OpenCourseWare.

2

Horiyama & Uehara 2010

regular octahedron & tetramonohedron

Image by MIT OpenCourseWare.

3

cube &tetramonohedron

Horiyama & Uehara 2010 Image by MIT OpenCourseWare.

4

regular icosahedron & 1 : 1.145 : 1.25 tetramonohedron

Horiyama & Uehara 2010

1

1

2 3

3 2

4 5

5

6

6

7 8 9 10

10987

4

Image by MIT OpenCourseWare.

5

Unfolding diagram removed due to copyright restrictions.

6

B

BD DCCA

A

E EFFGI

IJ

JHH

G

B

BD DCCA

A

E EFFGI

IJ

JHH

G

BB

DCCA

E EF F

II JH H

GJ

G

DA

Timothy Chan, 1999

unfoldings of √‾ × √‾ × 3 √‾ box2 2 2

unfolding of1 × 2 × 4 box

7

Therese Biedl, 1999

3 x 2 x 1

5 x 1 x 1

Image by MIT OpenCourseWare.

8

Therese Biedl, 1999

5 x 2 x 1

8 x 1 x 1

Image by MIT OpenCourseWare.

9

[Uehara 2008]Courtesy of Jun Mitani and Ryuhei Uehara. Used with permission.10

[Uehara 2008]Courtesy of Jun Mitani and Ryuhei Uehara. Used with permission.

11

common unfolding of 23 out of 24 pentacubes

[Aloupis, Benbernou, Bose, Collette, Demaine, Demaine, Douïeb, Dujmović, Iacono, Langerman, Morin 2010]

Image by MIT OpenCourseWare.

12

unique common unfolding of nonplanar pentacubes(& 22 total pentacubes)

[Aloupis, Benbernou, Bose, Collette, Demaine, Demaine, Douïeb, Dujmović, Iacono, Langerman, Morin 2010]

Image by MIT OpenCourseWare.

13

common unfolding of planar pentacubes(& no nonplanar pentacubes)

[Aloupis, Benbernou, Bose, Collette, Demaine, Demaine, Douïeb, Dujmović, Iacono, Langerman, Morin 2010]

Image by MIT OpenCourseWare.

14

Screenshot of applet showing pentacubes removed due to copyright restrictions.

15

[Donoso & O’Rourke

2002]

Image by MIT OpenCourseWare.See also http://arxiv.org/abs/cs/0110059/.

16

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

Tony Wills

S1 S2

Image by MIT OpenCourseWare.

18

[Benbernou, Cahn, O’Rourke 2004]

A

B

A’

Image by MIT OpenCourseWare.

19

Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.20

Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.

21

Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.22

Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.23

[Demaine, Demaine, Iacono, Langerman 2009]24

Courtesy of Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman. Used with permission.

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6.849 Geometric Folding Algorithms: Linkages, Origami, PolyhedraFall 2012

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