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:

    “Suppose we think while we talk or write—I mean, as we normally do—we shall not in general say that we think quicker than we talk, but the thought seems not to be separate from the expression.”
    —Ludwig Wittgenstein (1889–1951)

    “We have defined a story as a narrative of events arranged in their time-sequence. A plot is also a narrative of events, the emphasis falling on causality. “The king died and then the queen died” is a story. “The king died, and then the queen died of grief” is a plot. The time sequence is preserved, but the sense of causality overshadows it.”
    —E.M. (Edward Morgan)