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:

    “It was the words “descended into Hades”
    That seemed too pagan to our liberal youth.
    You know they suffered from a general onslaught.
    And well, if they weren’t true why keep right on
    Saying them like the heathen? We could drop them.”
    —Robert Frost (1874–1963)

    “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)