Abstract
In this paper we study the three-dimensional curves generated by the iterated bending of a piece of wire, which are generalizations of the so-called “dragon curves” or “paper-folding sequences” previously studied by Davis and Knuth, Mendès France, and other writers. These “wire-bending sequences” have several surprising properties. We characterize the nth term of a wire-bending sequence in terms of the binary expansion of n. We prove that the curves traced out in R 3 by many wire-bending sequences are actually bounded, although they are all aperiodic. Finally, we illustrate the close connection between wire-bending and the continued fractions for the transcendental numbers Σ n ⩾ 0 ε n g −2 n , where ε n = ± 1 and g ⩾ 3 is an integer.
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