Regular Paperfolding Sequence - General Paperfolding Sequence

General Paperfolding Sequence

The regular paperfolding sequence corresponds to folding a strip of paper consistently in the same direction. If we allow the direction of the fold to vary at each step we obtain a more general class of sequences. Given a binary sequence (f_i), we can define a general paperfolding sequence with folding instructions (f_i).

For a binary word w, let w‡ denote the reverse of the complement of w. Define an operator F_a as

and then define a sequence of words depending on the (f_i) by w₀ = ε,

The limit w of the sequence w_n is a paperfolding sequence. The regular paperfolding sequence corresponds to the folding sequence f_i = 1 for all i.

If n = m·2k where m is odd then

$t_n = \begin{cases} f_j & \text{if } m = 1 \mod 4 \\ 1-f_j & \text{if } m = 3 \mod 4 \end{cases}$

which may be used as a definition of a paperfolding sequence.

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