Part I: Linkages e: Unit-Length Locked Chains? Joseph ORourke Smith College.

Presentation on theme: "Part I: Linkages e: Unit-Length Locked Chains? Joseph ORourke Smith College."— Presentation transcript:

Part I: Linkages e: Unit-Length Locked Chains? Joseph ORourke Smith College

Open Problems: Locked Equilateral Chains? (1)Is there a configuration of a chain with universal joints, all of whose links have the same length, that is locked? (2)Is there a configuration of a 90 o fixed- angle chain, all of whose links have the same length, that is locked? Perhaps: No? Perhaps: Yes for 1+ ?

Minimal Stick Numbers zUpper Bounds for Equilateral Stick Numbers zEric J. Rawdon and Robert G. SchareinEric J. RawdonRobert G. Scharein zhttp://www.colab.sfu.ca/KnotPlot/ktheory.htmlhttp://www.colab.sfu.ca/KnotPlot/ktheory.html

The KnotPlot Site 831831 http://www.cs.ubc.ca/nest/imager/contributions/scharein/se/picts3.html

Open Problem: Cylindrical Equilateral Chains? (3)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]. ???

5 Unit-length Cylinders: Locked? [Sorina Chircu & JOR; 2000] [FUCG/PartI/cyls.nb]

Physical Model [Meghan Irving, Molly Miller, JOR; 2004]

Open Problem: Cylindrical Equilateral Chains? (4)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? (5) For a closed, unknotted chain (Martin Demain) ???