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

Part I: Linkages e: Unit-Length Locked Chains?. Joseph O’Rourke Smith College. 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?

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

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  1. Part I: Linkagese: Unit-Length Locked Chains? Joseph O’Rourke 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 1+e?

  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/scharein/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) ???

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