Hinged-rulers fold in $2-\Theta(\frac{1}{2^{n/4}})$

TL;DR

This paper proves that hinged-rulers with segments in [0,1] can always be folded into an interval slightly shorter than 2, and constructs rulers that cannot be folded into intervals significantly shorter than 2.

Contribution

It establishes tight bounds on the folding capabilities of hinged-rulers, showing both the maximum foldability and limitations for specific constructions.

Findings

Hinged-rulers can always be folded into an interval of length slightly less than 2.

There exist hinged-rulers that cannot be folded into an interval significantly shorter than 2.

The bounds are tight up to an exponential factor in the number of segments.

Abstract

A hinged-ruler is a sequence of line segments in the plane joined end-to-end with hinges, so each hinge joins exactly two segments, the first segment and last segment are adjacent to only one hinge each, and all other segments are adjacent to exactly two hinges. Hopcroft, Joseph, and Whitesides first posed the hinged-ruler-folding problem in their 1985 paper "On the Movement of Robot Arms in 2-Dimensional Bounded Regions": given a hinged-ruler and a real number $K$, can the hinged-ruler be folded so as to fit within a one-dimensional interval of length $K$? We show that if the segment lengths are constrained to be real numbers in the interval $[0,1]$, then we can always fold the hinged-ruler so as to fit within a one-dimensional interval of length $2-\Omega(\frac{1}{2^{n/4}})$. On the other hand, we give a construction for a hinged-ruler which cannot be folded into some one-dimensional interval of length $2-O(\frac{1}{2^{n/4}})$.