A Puzzle Paper Band

Abstract

PROF. C. V. BOYS'S puzzle (NATURE, June 9, p. 774) is a deal less puzzling (as he doubtless knows) if we begin it at the other end. Instead of making the long belt with its two loops which he describes, and then trying to reduce it to the well-known half-twist “double surface” (cf. e.g. Forsyth's “Differential Geometry,” p. 296) of double thickness, let us begin by laying two strips of paper one on the other; then with a half-twist bring the ends together, and fasten the corresponding ends each to each. Our half-twist will have brought one end of the lower strip into contact with the other end of the upper strip; and what we then obtain, on opening out, is the long loop (or “worble,” to use Maxwell's word) with its two curls, which Prof. Boys starts with. We have simply split into two sheets our original onesided, one-edged surface, and obtained a new bifacial surface thereby, precisely as Mr. B. M. Sen explains in his recent paper on “Double Surfaces” in the Proc. Lond. Math. Soc.