Folding Rulers inside Triangles

An l-ruler is a chain of n links, each of length l. The links, which are allowed to cross, are modeled by line segments whose endpoints act as joints. A given configuration of an l-ruler is said to fold if it can be moved to a configuration in which all its links coincide. We show that l-rulers confined inside an equilateral triangle of side 1 exhibit the following surprising alternation property: there are three values x1 ≈ 0.483, x2 = 0.5, and x3 ≈ 0.866 such that all configurations of n-link l-rulers fold if l ∈ [0, x1] or l ∈ (x2, x3], but, for any l ∈ (x1, x2] and any l ∈ (x3, 1], there are configurations of l-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.

• Type: Conference
• Year: 1993
• Where: CCCG
• Authors: Marc J. van Kreveld, Jack Snoeyink, Sue Whitesides
• Keywords: CCCG 1993, CCCG 2007, Length L, Line Segments, Surprising Alternation Property