From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: How do you translate that ? Date: 5 Oct 1999 22:02:02 GMT Newsgroups: sci.math Keywords: worm problem -- minimal convex set containing all length-1 curves In article , Jan Kok wrote: >> >> "What is the shortest curve (not necessarily closed) that does >> not fit in an equilateral triangle?" >> >Leaving aside the translation into French (to me the accents >in Axel's post obscured it to the extent that I stick to poor english) >what is known about the solution of this problem? > >With little effort I had a Z-shaped line of length (1 - (1/244)) that >cannot be enlarged. (I assume the equilateral triangle has sides of length 1.) > >But what about other shapes? From "Unsolved Problems in Geometry" (Kroft, Falconer, and Guy), section D18: The worm problem. Leo Moser asked what are the minimal comfortable living quarters for a "unit worm"? More precisely, it is required to find the convex set K of least area that contains a congruent copy of every continuous (rectifiable) curve of length 1. There are many possible formulations of this problem. We can summarize the problems by three "parameters", the first being the kind of equivalent copy that we are interested in (e.g. congruence or translation), the second being any restrictions on K, and the third any condition on the class of unit worms considered. [...] Besicovitch studied the problems [congruence, equilateral triangle, all worms] and [congruence, equilateral triangle, convex arcs], and in two papers, Wetzel discussed [tranlation/congruence, any triangle, closed worms], where the minimal triangles are equilateral, and [congruence, circular sections, all worms/closed worms]. Ref: A.S. Besicovitch, On arcs that cannot be covered by an open equilateral triangle of side 1, Math Gazette 49 (1965) 286-288; MR 32 #6320 J.E.Wetzel, Triangle covers for closed curves of constant length, Elemente Math. 25 (1970) 78-82 MR 42#960 J.E.Wetzel, On Moser's problem of accomodating closed curves in triangles, Elemente Math. 27 (1972) 35-36; MR 45 #4282 J.E.Wetzel, Sectorial covers for curves of constant length, Canad. Math. Bull. 16 (1973) 367-375, MR50#14451 The review of Besicovitch's article begins, "Let $\Lambda$ be the minimum length of an arc that cannot be covered by an open regular triangle of side 1. The author proves that $\Lambda\leq 0.98198$." dave