Abstract
Anl-ruler is a chain ofn links, each of lengthl. The links, which are allowed to cross, are modeled by line segments whose endpoints act as joints. A given configuration of anl-ruler is said to fold if it can be moved to a configuration in which all its links coincide. We show thatl-rulers confined inside an equilateral triangle of side 1 exhibit the following surprising alternation property: there are three valuesx 1≈0.483,x 2=0.5, andx 3≈0.866 such that all configurations ofn-linkl-rulers fold ifl∈[0,x 1] orl∈(x 2,x 3], but, for anyl∈(x 1,x 2] and anyl∈(x 3, 1], there are configurations ofl-rulers that cannot fold. In the folding cases, linear-time algorithms are given that achieve the folding. Also, a general proof technique is given that can show that certain configurations—in the nonfolding cases—cannot fold.
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Most of this research was done while the first author was at McGill University supported by an NSERC international fellowship. The second author was supported by NSERC. The third author was supported by NSERC and FCAR.
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van Kreveld, M., Snoeyink, J. & Whitesides, S. Folding rulers inside triangles. Discrete Comput Geom 15, 265–285 (1996). https://doi.org/10.1007/BF02711495
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DOI: https://doi.org/10.1007/BF02711495