Search

The mechanical procedure of paper folding generates an uncountable family of infinite sequences of fold patterns. We obtain the associated Fourier series and show that the sequences are almost periodic and hence deterministic. Further, we show that paper folding numbers defined by the sequences are all transcendental.

We study certain two-dimensional dynamical systems x n + 1 = F ( x n , y n ); y n + 1 = G ( x n , y n ).

Our remark sets up a correspondence ↔ between certain products of 2×2 matrices and continued fractions, which we shall exploit below. Of course this correspondence has an immediate geometric interpretation (cf Stark [1], Chap. 7). However, we shall obtain the usual properties of the continued fraction algorithm directly from the formalism rather than from the geometry. For example, we read that { pn+1 = cn+1pn + pn−1 qn+1 = cn+1qn + qn−1 n = 0, 1, 2, . . .