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A234434 Number of shapes of grid-filling curves (on the triangular grid) with turns by 0, +120, or -120 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns. 4
1, 1, 0, 0, 3, 0, 5, 0, 0, 10, 15, 0, 0, 17, 0, 0, 71, 0, 212, 0, 0, 0, 184, 0, 543, 842, 0, 0, 1848, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)



Shapes are considered modulo reflections and rotations.

The curves considered are not self-intersecting, not edge-contacting (i.e., have double edges), but (necessarily) vertex-contacting (i.e., a point in the grid is visited more than once).

The L-systems are interpreted as follows: 'F' is a unit-stroke in the current direction, '+' is a turn left by 120 degrees, '-' a turn right by 120 degrees, and '0' means "no turn".

The images in the links section use rounded corners to make the curves visually better apparent.

Three copies of each curve (connected by three turns '+' or three turns '-') give two tiles (that tile the triangular grid), but symmetric curves (any symmetry) give just one tile(-shape). The tiles are 3-symmetric, and sometimes (only for n of the form 6*k+1) 6-symmetric. There could in general be more tile-shapes than curve-shapes, for n=7 both cardinalities coincide, see links section. It turns out that for large n there are actually fewer tile-shapes than curve-shapes.

Terms a(n) are nonzero for n>=3 if and only if n is a term of A003136.

The equivalent sequence for the square grid has nonzero terms for n>=5 that are terms of A057653.

If more symbols are allowed for the L-systems, more curves are found, also if strokes of lengths other than one unit are allowed, see the Ventrella reference.


Table of n, a(n) for n=3..35.

Joerg Arndt, Matters Computational (The Fxtbook), see section 1.31.5 "Dragon curves based on radix-R counting", pp. 95-101, images of the R7-dragons are given on p. 97 and p. 98

Joerg Arndt, all 3 shapes of curves of order 7, rendered after 4 generations of the L-systems.

Joerg Arndt, all 3 shapes of tiles of order 7, rendered after 4 generations of the L-systems, curves colored to make them apparent.

Joerg Arndt, all 15 shapes of curves of order 13, rendered after 3 generations of the L-systems (file size about 500 kB).

Joerg Arndt, all shapes of tiles of order 13, rendered after 3 generations of the L-systems (file size about 500 kB). Note: not all symmetries are accounted for, so some tiles appear more than once (e.g., in flipped over form).

Joerg Arndt, decompositions of order-13 curves into self-similar parts (file size about 1.3 MB)

Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.

Jeffrey J. Ventrella, Brain-Filling Curves: A Fractal Bestiary, 2012.


The a(3)=1 shape of order 3 is generated by F |--> F+F-F, the curve generated by F |--> F-F+F has the same shape (after reflection). The curve is called the "terdragon", see A080846.

There are 5 L-systems that generate a curve of order 7 with first turn '0' or '+':

F |--> F0F+F0F-F-F+F  #  R7-1

F |--> F0F+F+F-F-F0F  #  R7-2

F |--> F+F0F+F-F0F-F  #  R7-3

F |--> F+F-F-F0F+F0F  #  R7-4 # same shape as R7-1

F |--> F+F-F-F+F+F-F  #  R7-5 # same shape as R7-2

As shown, these give just 3 shapes (and the L-systems with first turn '-' give no new shapes), so a(7)=3.

The curve R7-1 appears on page 107 in the Ventrella reference.

The symmetric curves R7-2 and R7-5 appear in the Arndt reference (there named "R7-dragon" and "second R7-dragon", see A176405 and A176416).


Cf. A265685 (shapes on the square grid), A265686 (tri-hexagonal grid).

Sequence in context: A227498 A131986 A002656 * A234020 A276833 A166586

Adjacent sequences:  A234431 A234432 A234433 * A234435 A234436 A234437




Joerg Arndt, Dec 26 2013



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