Part I: Linkages e: Unit-Length Locked Chains?

1
Part I Linkagese Unit-Length Locked Chains?

• Joseph ORourke
• Smith College

2
Open Problems Locked Equilateral Chains?

• Is there a configuration of a chain with
  universal joints, all of whose links have the
  same length, that is locked?
• Is there a configuration of a 90o fixed-angle
  chain, all of whose links have the same length,
  that is locked?

Perhaps No?
Perhaps Yes for 1e?
3
Minimal Stick Numbers

• Upper Bounds for Equilateral Stick Numbers
• Eric J. Rawdon and Robert G. Scharein
• http//www.colab.sfu.ca/KnotPlot/ktheory.html

4
The KnotPlot Site
831
http//www.cs.ubc.ca/nest/imager/contributions/sch
arein/se/picts3.html
5
Open Problem Cylindrical Equilateral Chains?

• Is there a configuration of a chain of five
  cylinders with universal joints, all of whose
  links have the same length, that is locked?
• (Five links of different length can be locked
• five links of unit length cannot be locked.
  Cantarella Johnson, 1998.

???
6
5 Unit-length Cylinders Locked?
Sorina Chircu JOR 2000
FUCG/PartI/cyls.nb
7
Physical Model
Meghan Irving, Molly Miller, JOR 2004
8
Open Problem Cylindrical Equilateral Chains?

• Is there a configuration of a chain of any number
  of cylinders with universal joints, all of whose
  links have the same length, that is locked?
• For a closed, unknotted chain (Martin Demain)

???