Polygonal Chains Cannot Lock in 4D
Abstract
We prove that, in all dimensions d>=4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polynomial number of ``moves.'' These results contrast to those known for d=2, where trees can ``lock,'' and for d=3, where open and closed chains can lock.
Keywords
Cite
@article{arxiv.cs/9908005,
title = {Polygonal Chains Cannot Lock in 4D},
author = {Roxana Cocan and Joseph O'Rourke},
journal= {arXiv preprint arXiv:cs/9908005},
year = {2007}
}
Comments
Major revision of the Aug. 1999 version, including: Proof extended to show trees cannot lock in 4D; new example of the implementation straightening a chain of n=100 vertices; improved time complexity for chain to O(n^2); fixed several minor technical errors. (Thanks to three referees.) 29 pages; 15 figures. v3: Reference updated