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A265671 Directions of edges in a plane-filling curve of order 13. 1


%S 1,2,2,2,1,3,1,1,2,3,2,2,1,2,3,3,3,2,1,2,2,3,1,3,3,2,2,3,3,3,2,1,2,2,

%T 3,1,3,3,2,2,3,3,3,2,1,2,2,3,1,3,3,2,1,2,2,2,1,3,1,1,2,3,2,2,1,3,1,1,

%U 1,3,2,3,3,1,2,1,1,3,1,2,2,2,1,3,1,1,2,3,2,2,1,1,2,2,2,1,3,1,1,2,3,2,2,1,2,3,3

%N Directions of edges in a plane-filling curve of order 13.

%C Infinite ternary word generated from the axiom 1 by the Lindenmayer system with maps 1 --> 1222131123221, 2 --> 2333212231332, and 3 --> 3111323312113.

%C This is a 13-automatic sequence. It can be generated by reading the lowest nonzero digit D in the base-13 expansion of n>=1: a(n)=1 for D \in {1, 5, 7, 8}, a(n)=2 for D \in {2, 3, 4, 9, 11, 12}, and a(n)=3 for D \in {6, 10}.

%C Corresponds to a grid-filling curve on the triangular grid as a sequence of directed edges where the letters are the directions of the third roots of unity. See the file titled "First iterate of the curve".

%C The corresponding sequence of turns (by 0 or +-120 degree) can be obtained from the L-system with axiom + and maps + --> +00--+0++-0-+, 0 --> +00--+0++-0-0, and - --> +00--+0++-0--.

%C The shape of the curve is one of the A234434(13)=15 possible shapes.

%C An L-system with axiom F and just one non-constant map F --> F+F0F0F-F-F+F0F+F+F-F0F-F generates the curve when 0, +, and - are interpreted as turns and F as a unit stroke in the current direction.

%C Three copies of the curve can be arranged to create a rep-tile that is a lattice tiling, see the files "Tile-plus" (axiom F+F+F), "Tile-minus" (Axiom F-F-F), "Tiling-plus" (self-similarity of the Tile-plus), and "Complex numeration system" (giving the generalized unit square of a numeration system with base 1 + i * sqrt(12) that reproduces the Tile-plus).

%H Joerg Arndt, <a href="/A265671/b265671.txt">Table of n, a(n) for n = 1..2197</a>

%H Joerg Arndt, <a href="/A265671/a265671.pdf">First iterate of the curve</a>, <a href="/A265671/a265671_1.pdf">Second iterate</a>, <a href="/A265671/a265671_2.pdf">Third iterate</a>, <a href="/A265671/a265671_3.pdf">Fourth iterate</a>.

%H Joerg Arndt, <a href="/A265671/a265671.png">Rendering used for the T-shirt on Neil's 75th birthday</a> (png image, 1716 X 2732 pixel).

%H Joerg Arndt, <a href="/A265671/a265671_4.pdf">Tile-plus</a>, <a href="/A265671/a265671_5.pdf">Tile-minus</a>, <a href="/A265671/a265671_6.pdf">Tiling-plus</a>, <a href="/A265671/a265671_1.png">Complex numeration system</a>.

%H Joerg Arndt, <a href="http://arxiv.org/abs/1607.02433">Plane-filling curves on all uniform grids</a>, arXiv:1607.02433 [math.CO], (8-July-2016)

%H <a href="/index/Ar#13-automatic">Index entries for 13-automatic sequences</a>.

%Y Cf. A029883, A035263, A060236, A080846, A156595, A175337, A176405, A176416.

%Y Cf. A234434 (curves on the triangular grid).

%Y Cf. A229214 (a similar L-system for Gosper's flowsnake).

%K nonn

%O 1,2

%A _Joerg Arndt_, Dec 13 2015

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