OFFSET
1,5
COMMENTS
Curves of order n generated by folding morphisms are walks on the square grid, also coded by sequences (starting with D) of n-1 U's and D's, the Up and Down folds. These are also known as n-folds. In the square grid they uniquely correspond to folding morphisms, which are a special class of morphisms sigma on the alphabet {a,b,c,d}. (There is in particular the requirement that the first two letters of sigma(a) are ab.) Here the letters a,b,c, and d correspond to the four possible steps of the walk.
A curve C = C1 of order n generates curves Cj of order n^j by the process of iterated folding. Iterated folding corresponds to iterates of the folding morphism. Grid-filling or plane-filling means that all the points in arbitrary large balls of gridpoints are eventually visited by the Cj.
REFERENCES
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted and updated in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614. See page 611, table A_s = a(s).
LINKS
Michel Dekking, Table of n, a(n) for n = 1..20
F. M. Dekking, Paperfolding Morphisms, Planefilling Curves, and Fractal Tiles, Theoretical Computer Science, volume 414, issue 1, January 2012, pages 20-37. Also arXiv:1011.5788 [math.CO], 2010-2011.
EXAMPLE
Examples for n = 5 are given in Knuth's 2010 update. There are pictures which show (or suggest) that the 5-folds coded by DDUU, DUDD, DDUD are perfect, DUUD and DUDU yield a self-avoiding curve which is not plane-filling, and the other 3 give self-intersecting curves. So A343992(5) = 3 and a(5) = 2.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, May 06 2021
EXTENSIONS
Rewritten and renamed by Michel Dekking, Jun 06 2021
STATUS
approved