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
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 has been the struggle between privileged men who have managed to get hold of the levers of power and the people in general with their vague and changing aspirations for equality, for justice, for some kind of gentler brotherhood and peace, which has kept that balance of forces we call our system of government in equilibrium.”
—John Dos Passos (1896–1970)
“Reminiscences, even extensive ones, do not always amount to an autobiography.... For autobiography has to do with time, with sequence and what makes up the continuous flow of life. Here, I am talking of a space, of moments and discontinuities. For even if months and years appear here, it is in the form they have in the moment of recollection. This strange form—it may be called fleeting or eternal—is in neither case the stuff that life is made of.”
—Walter Benjamin (1892–1940)