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

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, 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, 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

Offset: 0

Author

Arie Bos, Jul 27 2022

Keywords

• sign

Comments

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.

Dekking's "Recurrent sets" published this first, but this "E-curve" was discovered in 1978 by Douglas McKenna.

References

• 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.

Links

• Arie Bos, Fractal Images as Number Sequences I, arXiv:2207.12942 [cs.CG], 2022. See sec. 3.8, appendix B3.
• F. M. Dekking, Recurrent sets, Adv. Math., 44 (1982), 78-104. See example 4.9.

Crossrefs

Other curves: A229214, A261180.

Formula

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

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)),

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.

(See Bos arXiv link, appendix B3.)