1
This image is in the public domain.
hexokinase (PDB 1QHA)
drawn by Tim Vickers
2
2
1
4
8
6
3
Image by MIT OpenCourseWare.
minimum flat span is NP-complete
Image by MIT OpenCourseWare.
maximum flat span is NP-complete
3
[Soss 2001]
v3
v0
e’4
e’1
v4
v3
v2
v1
v0
e’1
e’4
v2
v1
v4
Image by MIT OpenCourseWare.
4
en
l
σ
l
x
σ
l
Image by MIT OpenCourseWare.
5
[Soss & Toussaint 2000]
CONSTRAINTS ON FIXED-ANGLE LINKAGE
Connectivity
Open chain
Chain Geometry
Flat-state connectivity
Has monotone state
Connected
α orthogonal or obtuse
Connected
α equal acute
Connected
Unit; α
Connected
(60º, 150º)
Multiple pinned open chains Orthogonal
Connected
Closed chain
Unit; orthogonal
Connected
Tree
Orthogonal; partially rigid
Disconnected
Graph
Orthogonal
Disconnected
“Unit” means with unit link lengths; a “monotone state” is a planner, strictly monotone configuration; α represents the joint angles.
Image by MIT OpenCourseWare.
6
c1
b
a3
c
d1
b1
a
x
b1
b
c
a3
d
x
d
a1
a1
a
d1
c1
Image by MIT OpenCourseWare.
7
[Aloupis, Demaine, Dujmović, Erickson,
Langerman, Meijer, O’Rourke, Overmars,
Soss, Streinu, Toussaint 2002]
c1
b
a3
c
d1
b1
a
x
b
a
c
x
d
d
a1
Images by MIT OpenCourseWare.
8
[Aloupis, Demaine, Dujmović, Erickson,
Langerman, Meijer, O’Rourke, Overmars,
Soss, Streinu, Toussaint 2002]
[Aloupis,
Demaine,
Dujmović,
Erickson,
Langerman,
Meijer,
O’Rourke,
Overmars,
Soss,
Streinu,
Toussaint
2002]
c1
c2
a2
b
c
a
a1
9
a2
a
b1
b2
Images by MIT OpenCourseWare.
(A)
a3
b
a
R
b3
S
(B)
a3
b
R
b3
a
S
Image by MIT OpenCourseWare.
10
[Aloupis, Demaine, Dujmović,
Erickson, Langerman, Meijer,
O’Rourke, Overmars, Soss,
Streinu, Toussaint 2002]
a
c
v0
b
v2
v1
e3
e4
v3
Image by MIT OpenCourseWare.
11
[Aloupis, Demaine, Dujmović, Erickson,
Langerman, Meijer, O’Rourke, Overmars,
Soss, Streinu, Toussaint 2002]
f
d
e
v4
v3
Image by MIT OpenCourseWare.
12
[Aloupis, Demaine, Dujmović, Erickson,
Langerman, Meijer, O’Rourke, Overmars,
Soss, Streinu, Toussaint 2002]
Date: Sat, 11 May 2002 09:37:05
From: 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 Email
answer
is no because the crossed leg
from Stefan Langerman (2002) removed due to copyright restrictions.
config can be made unit length:
13
*
/|
-----------/------*
/ |
/
/
|
/
/
| /
/
| /
/
| /
|/
|/
*
Courtesy of Stefan Langerman. Used with permission.
[Langerman 2002]
14
Image by MIT OpenCourseWare.
15
Fig. 1 removed due to copyright restrictions.
Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on Producible
Polygonal Protein Chains." $OJRULWKPLFD 44, no. 2 (2006): 167–81.
16
[Demaine, Langerman,
O’Rourke 2006]
Fig. 2 removed due to copyright restrictions.
Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on Producible
Polygonal Protein Chains." $OJRULWKPLFD 44, no. 2 (2006): 167–81.
17
[Demaine, Langerman,
O’Rourke 2006]
Fig. 3 removed due to copyright restrictions.
Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on Producible
Polygonal Protein Chains." $OJRULWKPLFD 44, no. 2 (2006): 167–81.
18
[Demaine, Langerman,
O’Rourke 2006]
Fig. 5 removed due to copyright restrictions.
Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on Producible
Polygonal Protein Chains." $OJRULWKPLFD 44, no. 2 (2006): 167–81.
19
[Demaine, Langerman,
O’Rourke 2006]
climber plant
in Costa Rica
Fig. 5 removed due to copyright restrictions.
Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on Producible
Polygonal Protein Chains." $OJRULWKPLFD 44, no. 2 (2006): 167–81.
Photograph of climber plant removed due to copyright restrictions.
[Demaine,
Langerman,
O’Rourke
2006]
20
Fig. 8 removed due to copyright restrictions.
Refer to: Demaine, E. D., S. Langerman, et al. "Geometric Restrictions on Producible
Polygonal Protein Chains." $OJRULWKPLFD 44, no. 2 (2006): 167–81.
[Demaine, Langerman,
O’Rourke 2006]
21
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6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra
Fall 2012
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