hexokinase (PDB 1QHA)drawn by Tim VickersThis image is in the public domain.1

2

minimum flat span is NP-complete

maximum flat span is NP-complete

[Soss 2001]

1 8

24

63

Image by MIT OpenCourseWare.

Image by MIT OpenCourseWare.

3

e’4

v1

v4

v2

v3

v0

e’1

e’4

v1v4

v2

v3

v0

e’1

Image by MIT OpenCourseWare.

4

[Soss & Toussaint 2000]

en

xll lσ σ

Image by MIT OpenCourseWare.

5

Image by MIT OpenCourseWare.

6

CONSTRAINTS ON FIXED-ANGLE LINKAGE

Flat-state connectivity

Connected

Connected

Connected

Connected

Connected

Connected

Open chain

Multiple pinned open chains

Closed chain

Tree

Graph

“Unit” means with unit link lengths; a “monotone state” is a planner, strictly monotone configuration; α represents the joint angles.

Has monotone state

α orthogonal or obtuse

α equal acute

Unit; α (60º, 150º)

Orthogonal

Unit; orthogonal

Orthogonal; partially rigid

Orthogonal

DisconnectedDisconnected

Connectivity Chain Geometry

[Aloupis, Demaine, Dujmović, Erickson, Langerman, Meijer, O’Rourke, Overmars,

Soss, Streinu, Toussaint 2002]

c a

d

b b1

a1

d1

a3

c1

xc a

d

bb1

a1

c1

d1

a3 x

Image by MIT OpenCourseWare.

7

[Aloupis, Demaine, Dujmović, Erickson, Langerman, Meijer, O’Rourke, Overmars,

Soss, Streinu, Toussaint 2002]

b

c x

d

ac a

d

b b1

a1

d1

a3

c1

x

Images by MIT OpenCourseWare.

8

[Aloupis, Demaine,

Dujmović, Erickson,

Langerman, Meijer,

O’Rourke, Overmars,

Soss, Streinu,

Toussaint 2002]

a1

a2

c1

c

c2

a

ba2

b1

b2

a

Images by MIT OpenCourseWare.9

[Aloupis, Demaine, Dujmović, Erickson, Langerman, Meijer,

O’Rourke, Overmars, Soss, Streinu, Toussaint 2002]

a3

(A)

(B) a3

b3R

S

S

b3R a

a

b

b

Image by MIT OpenCourseWare.

10

[Aloupis, Demaine, Dujmović, Erickson, Langerman, Meijer, O’Rourke, Overmars,

Soss, Streinu, Toussaint 2002]

c

a

b

v0

v2v1

v3

e4e3

Image by MIT OpenCourseWare.

11

[Aloupis, Demaine, Dujmović, Erickson, Langerman, Meijer, O’Rourke, Overmars,

Soss, Streinu, Toussaint 2002]

e df

v3

v4

Image by MIT OpenCourseWare.

12

Date: Sat, 11 May 2002 09:37:05From: Stefan Langerman <sl@jeff.cs.mcgill.ca>To: Joseph O'Rourke <orourke@turing.csc.smith.edu>Cc: Erik Demaine <edemaine@mit.edu>Subject: Re: Molecule machine: questions[…]Q3a. Is every unit-length embedded chain flattenable?I think the answer is no because the crossed leg config can be made unit length:

*/|

-----------/------*/ | /

/ | // | /

/ | // | /

|/|/*

Courtesy of Stefan Langerman. Used with permission.13

Email from Stefan Langerman (2002) removed due to copyright restrictions.

[Langerman 2002]14

Image by MIT OpenCourseWare.

15

16

Fig. 1 removed due to copyright restrictions. Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on ProduciblePolygonal Protein Chains." Algorithmica 44, no. 2 (2006): 167–81.

[Demaine, Langerman, O’Rourke 2006]

17

Fig. 2 removed due to copyright restrictions. Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on ProduciblePolygonal Protein Chains." Algorithmica 44, no. 2 (2006): 167–81.

[Demaine, Langerman, O’Rourke 2006]

[Demaine, Langerman, O’Rourke 2006]18

Fig. 3 removed due to copyright restrictions. Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on ProduciblePolygonal Protein Chains." Algorithmica 44, no. 2 (2006): 167–81.

[Demaine, Langerman, O’Rourke 2006]

19

Fig. 5 removed due to copyright restrictions. Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on ProduciblePolygonal Protein Chains." Algorithmica 44, no. 2 (2006): 167–81.

[Demaine, Langerman,

O’Rourke 2006]

climber plant in Costa Rica

Photograph of climber plant removed due to copyright restrictions.

20

Fig. 5 removed due to copyright restrictions. Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on ProduciblePolygonal Protein Chains." Algorithmica 44, no. 2 (2006): 167–81.

[Demaine, Langerman, O’Rourke 2006]

21

Fig. 8 removed due to copyright restrictions. Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on ProduciblePolygonal Protein Chains." Algorithmica 44, no. 2 (2006): 167–81.

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6.849 Geometric Folding Algorithms: Linkages, Origami, PolyhedraFall 2012

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