cp's OEIS Frontend

A356112 Direction of segment n in the E curve of Dekking and McKenna.

%I A356112 #42 Jul 30 2022 19:39:40
%S A356112 1,1,2,-1,2,1,2,-1,-1,2,1,1,1,2,1,-2,-2,-1,-2,-2,1,2,1,-2,-2,1,1,2,-1,
%T A356112 2,1,2,-1,-1,2,1,1,1,2,1,-2,-2,-1,-2,-2,1,2,1,-2,-2,1,1,2,-1,2,1,1,-2,
%U A356112 1,1,2,-1,2,2,2,-1,-2,-2,-1,2,-1,-2,-1,2,2,2,2,-1,-2,-1,2,2,1,2,2,-1,-2,-1,-1,-1,-2,1,1,-2,-1,-2,1,-2,-1,-1,2,2,-1,-2
%N A356112 Direction of segment n in the E curve of Dekking and McKenna.
%C A356112 On the square grid go one step to the left for a -1, one to the right for a +1, one down for a -2, and one up for a +2. Otherwise stated, replace +-1 with the vector +-(1,0) and +-2 with the vector +-(0,1), then take the running sum to obtain all the vertices of the fractal.
%C A356112 Dekking's "Recurrent sets" published this first, but this "E-curve" was discovered in 1978 by Douglas McKenna.
%D A356112 Douglas M. McKenna, "SquaRecurves, E-Tours, Eddies, and Frenzies: Basic Families of Peano Curves on the Square Grid", in "The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and its History", Mathematical Association of America, 1994, pages 49-73, ISBN 0-88385-516-X.
%H A356112 Arie Bos, <a href="https://arxiv.org/abs/2207.12942">Fractal Images as Number Sequences I</a>, arXiv:2207.12942 [cs.CG], 2022. See sec. 3.8, appendix B3.
%H A356112 F. M. Dekking, <a href="https://doi.org/10.1016/0001-8708(82)90066-4">Recurrent sets</a>, Adv. Math., 44 (1982), 78-104. See example 4.9.
%F A356112 If s=[a,b] is a signed permutation, then s(1)=a, s(2)=b, s(-x)=-s(x), a,b,x in {1,2,-1,-2}. Substitution T is defined by T(i) = (i, i, ut, -t, u,   i, ut, -t, -i, ut,   i, i, t, u, t,   -u, -u, -t, -u, -ut,   t, u, i, -ut, -ut), where the signed permutations are defined by i=[1,2], t=[1, -2], u=[2, -1]. The start of the substitution is 1. This means that
%F A356112 T([1,2]x)=([1,2](x),  [1,2](x),  [2,-1][1,-2](x),  -[1,-2](x),  [2,-1](x),     [1,2](x),  [2,-1][1,-2](x),  -[1,-2](x),  -[1,2](x),  [2,-1][1,-2](x),     [1,2](x),  [1,2](x),  [1,-2](x),  [2,-1](x),  [1,-2](x),     -[2,-1](x),  -[2,-1](x),  -[1,-2](x),  -[2,-1](x),  -[2,-1][1,-2](x),     [1,-2](x),  [2,-1](x),  [1,2](x),  -[2,-1][1,-2](x),  -[2,-1][1,-2])(x)),
%F A356112 So T(1)=(1,1,2,-1,2,  1,2,-1,-1,2,  1,1,1,2,1,  -2,-2,-1,-2,-2,  1,2,1,-2,-2) etc.
%F A356112 (See Bos arXiv link, appendix B3.)
%Y A356112 Other curves: A229214, A261180.
%K A356112 sign
%O A356112 0,3
%A A356112 _Arie Bos_, Jul 27 2022