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

This paper investigates the existence of locked chains composed of equilateral links and cylindrical links with universal joints, each link having the same length. We pose open questions regarding configurations, particularly for equilateral chains that are fixed at a 90-degree angle, and cylindrical chains with varied lengths. Recent findings suggest potential answers for configurations with five links, yet conclusive results remain elusive. This research connects to advanced geometric studies and contributes to the field of knot theory and robotics.

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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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