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 (fi), we can define a general paperfolding sequence with folding instructions (fi).

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

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

The limit w of the sequence wn is a paperfolding sequence. The regular paperfolding sequence corresponds to the folding sequence fi = 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.

Read more about this topic:  Regular Paperfolding Sequence

Famous quotes containing the words general and/or sequence:

    “As a general truth, it is safe to say that any picture that produces a moral impression is a bad picture.”
    —Edmond De Goncourt (1822–1896)

    “It isn’t that you subordinate your ideas to the force of the facts in autobiography but that you construct a sequence of stories to bind up the facts with a persuasive hypothesis that unravels your history’s meaning.”
    —Philip Roth (b. 1933)