Properties
The value of any given term tn in the regular paperfolding sequence can be found recursively as follows. If n = m·2k where m is odd then
Thus t12 = t3 = 0 but t13 = 1.
The paperfolding word 1101100111001001..., which is created by concatenating the terms of the regular paperfolding sequence, is a fixed point of the morphism or string substitution rules
- 11 → 1101
- 01 → 1001
- 10 → 1100
- 00 → 1000
as follows:
- 11 → 1101 → 11011001 → 1101100111001001 → 11011001110010011101100011001001 ...
It can be seen from the morphism rules that the paperfolding word contains at most three consecutive 0s and at most three consecutive 1s.
The paperfolding sequence also satisfies the symmetry relation:
which shows that the paperfolding word can be constructed as the limit of another iterated process as follows:
- 1
- 1 1 0
- 110 1 100
- 1101100 1 1100100
- 110110011100100 1 110110001100100
In each iteration of this process, a 1 is placed at the end of the previous iteration's string, then this string is repeated in reverse order, replacing 0 by 1 and vice versa.
Read more about this topic: Regular Paperfolding Sequence
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