Moser's Worm Problem
Find the smallest-area planar region that can cover every curve of length 1.
Moser's worm problem asks for the region of smallest area that can accommodate every plane curve of unit length, where each curve may be rotated and translated to fit. The problem is visually natural but technically subtle because every possible bent, kinked, or smoothly curved arc of length one must fit inside a single compact region. The best known non-convex upper bound of approximately 0.260437 was established by Norwood and Poole in 2003, while the best convex lower bound of 0.232239 was proved by Khandhawit, Sriswasdi, and Wetzel in 2013. Despite steady progress on tightening bounds, the exact minimum area remains unknown.
Formula
Find the convex region of smallest area that can contain a congruent copy of every plane curve of unit length.
The optimal area lies between Khandhawit–Pagonakis (2014) lower and Norwood–Poole (2003) upper bounds — still a wide gap.
A semicircle of diameter 1 is a valid universal cover — far from optimal but a natural starting point; trimming corners improves it.
Summary
Moser's worm problem asks for the region of smallest area that can accommodate every plane curve of unit length, where each curve may be rotated and translated to fit. The problem is visually natural but technically subtle because every possible bent, kinked, or smoothly curved arc of length one must fit inside a single compact region. The best known non-convex upper bound of approximately 0.260437 was established by Norwood and Poole in 2003, while the best convex lower bound of 0.232239 was proved by Khandhawit, Sriswasdi, and Wetzel in 2013. Despite steady progress on tightening bounds, the exact minimum area remains unknown.
