TejasviTomar
October 2026 · Preprint

A lower bound for Moser's worm problem with rotations and translations

Tejasvi Singh Tomar

In plain words

Moser's worm problem asks for the smallest region that can hold a copy of every curve of length 1. This paper considers regions without dents, in which the copies may be turned and slid but not flipped. The best lower bound for their area was 0.239, which Keller proved in a 2026 preprint for the version that also allows flipping. Without flipping, the paper raises it to 0.2405, using five curves made of straight pieces. A computer search over all their positions produced a certificate of 203,209,418 cases, which a separate program checks with rigorous error bounds.

Abstract

Moser's worm problem asks for the least area of a convex set in the plane that contains a copy of every arc of length 1. We consider the version in which the copies may be rotated and translated but not reflected, and give a computer-assisted proof that the least area M+ satisfies M+ ≥ 0.2405. The previous lower bound, 0.239, was proved by Keller for the version with reflections and carries over to this one. Like the earlier lower bounds, ours uses that such a set contains the convex hull of copies of finitely many arcs. We take five polygonal arcs: a segment, an equilateral two-link arc, a three-link arc and two zigzag arcs that turn in opposite directions. Without reflections, a placed zigzag can serve only as a copy of the zigzags that turn in its own direction. On a box of placements we bound the hull area from below by signed areas of polygons spanned by vertices of the placed arcs. We bound these areas by second-order Taylor expansions with certified eigenvalue bounds and, where the hull changes its combinatorial type, by convex combinations of such bounds. Our branch and bound over a twelve-dimensional box of placements produced a certificate with 203,209,418 leaves, which a separate checker verifies in ball arithmetic.

Top: a wide, low, light teal region shaped like a flattened triangle, outlined in thin teal, with a black horizontal segment along its base. Inside it are a teal inverted V reaching the apex, a magenta arc shaped like an open box, and two long crossing zigzags in yellow and green that span most of its width. Bottom: the same five arcs separately and smaller, in a row: the black segment, the teal inverted V over a shaded triangle, the magenta arc over a shaded quadrilateral, and the yellow and green zigzags over shaded slanted parallelograms. No axes or labels.
Illustration of the five arcs of the proof, each of length 1: a segment (black), a two-link arc whose links are two sides of an equilateral triangle (teal), a three-link arc (magenta), and two zigzags that turn in opposite directions (yellow and green). Bottom: the five arcs side by side, each with its convex hull shaded. Top, at about three times that scale: a placement of the arcs by rotations and translations, with no reflections, whose convex hull (shaded) has area 0.24205381…, the smallest found by local minimization from 300 random starting points. By the paper's theorem, every convex set that holds a rotated and translated copy of each of the five arcs has area at least 0.2405, so the least area of such a set is at least 0.2405 and at most this hull's area. Without reflections, neither zigzag can be flipped to turn like the other, so the two constrain the hull separately. The placement is the one recorded in data/opt_point.json of the repository; its hull area, recomputed here from the exact arcs in floating point, agrees with the recorded value.