Math::PlanePath::SierpinskiArrowhead -- self-similar triangular path traversal
use Math::PlanePath::SierpinskiArrowhead; my $path = Math::PlanePath::SierpinskiArrowhead->new; my ($x, $y) = $path->n_to_xy (123);
This path is an integer version of Sierpinski's curve from
Waclaw Sierpinski, "Sur une Courbe Dont Tout Point est un Point de Ramification", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, volume 160, January-June 1915, pages 302-305. http://gallica.bnf.fr/ark:/12148/bpt6k31131/f302.image.langEN
The path is self-similar triangular parts leaving middle triangle gaps giving the Sierpinski triangle shape.
\ 27----26 19----18 15----14 8 \ / \ / \ 25 20 17----16 13 7 / \ / 24 21 11----12 6 \ / / 23----22 10 5 \ 5---- 6 9 4 / \ / 4 7---- 8 3 \ 3---- 2 2 \ 1 1 / 0 <- Y=0 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8
The base figure is the N=0 to N=3 shape. It's repeated up in mirror image as N=3 to N=6 then across as N=6 to N=9. At the next level the same is done with the N=0 to N=9 shape, up as N=9 to N=18 and across as N=18 to N=27, etc.
The X,Y coordinates are on a triangular lattice done in integers by using every second X, per "Triangular Lattice" in Math::PlanePath.
The base pattern is a triangle like
3---------2 - - - - . \ \ C / \ B / \ D \ / \ / . - - - - 1 \ / A / \ / / 0
Higher levels go into the triangles A,B,C but the middle triangle D is not traversed. It's hard to see that omitted middle in the initial N=0 to N=27 above. The following is more of the visited points, making it clearer
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
The path is related to the Sierpinski triangle or "gasket" by treating each line segment as the side of a little triangle. The N=0 to N=1 segment has a triangle on the left, N=1 to N=2 on the right, and N=2 to N=3 underneath, which are per the A,B,C parts shown above. Notice there's no middle little triangle "D" in the triplets of line segments. In general a segment N to N+1 has its little triangle to the left if N even or to the right if N odd.
This pattern of little triangles is why the N=4 to N=5 looks like it hasn't visited the vertex of the triangular N=0 to N=9 -- the 4 to 5 segment is standing in for a little triangle to the left of that segment. Similarly N=13 to N=14 and each alternate side midway through replication levels.
There's easier ways to generate the Sierpinski triangle though. One of the simplest is to take X,Y coordinates which have no 1 bit on common, ie. a bitwise-AND,
($x & $y) == 0
which gives the shape in the first quadrant X>=0,Y>=0. The same can be had with the ZOrderCurve path by plotting all numbers N which have no digit 3 in their base-4 representation (see "Power of 2 Values" in Math::PlanePath::ZOrderCurve), since digit 3s in that case are X,Y points with a 1 bit in common.
ZOrderCurve
The attraction of this Arrowhead path is that it makes a connected traversal through the Sierpinski triangle pattern.
Counting the N=0,1,2,3 part as level 1, each level goes from
Nstart = 0 Nlevel = 3^level
inclusive of the final triangle corner position. For example level 2 is from N=0 to N=3^2=9. Each level doubles in size,
0 <= Y <= 2^level - 2^level <= X <= 2^level
The final Nlevel position is alternately on the right or left,
Xlevel = / 2^level if level even \ - 2^level if level odd
The Y axis is crossed, ie. X=0, at N=2,6,18,etc which is is 2/3 through the level, ie. after two replications of the previous level,
Ncross = 2/3 * 3^level = 2 * 3^(level-1)
An optional align parameter controls how the points are arranged relative to the Y axis. The default shown above is "triangular". The choices are the same as for the SierpinskiTriangle path.
align
SierpinskiTriangle
"right" means points to the right of the axis, packed next to each other and so using an eighth of the plane.
align => "right" | | 8 | 27-26 19-18 15-14 | | / | / | 7 | 25 20 17-16 13 | / | / 6 | 24 21 11-12 | | / / 5 | 23-22 10 | | 4 | 5--6 9 | / | / 3 | 4 7--8 | | 2 | 3--2 | | 1 | 1 | / Y=0 | 0 +-------------------------- X=0 1 2 3 4 5 6 7
"left" is similar but skewed to the left of the Y axis, ie. into negative X.
align => "left" \ 27-26 19-18 15-14 | 8 \ | \ | \ | 25 20 17-16 13 | 7 | \ | | 24 21 11-12 | 6 \ | | | 23-22 10 | 5 \ | 5--6 9 | 4 | \ | | 4 7--8 | 3 \ | 3--2 | 2 \ | 1 | 1 | | 0 | Y=0 -----------------------------+ -8 -7 -6 -5 -4 -3 -2 -1 X=0
"diagonal" put rows on diagonals down from the Y axis to the X axis. This uses the whole of the first quadrant (with gaps).
align => "diagonal" | | 8 | 27 | \ 7 | 26 | | 6 | 24-25 | | 5 | 23 20-19 | \ | \ 4 | 22-21 18 | | 3 | 4--5 17 | | \ \ 2 | 3 6 16-15 | \ | \ 1 | 2 7 10-11 14 | | \ | \ | Y=0 | 0--1 8--9 12-13 +-------------------------- X=0 1 2 3 4 5 6 7
Sierpinski presents the curve with a base along the X axis. That can be had here with a -60 degree rotation (see "Triangular Lattice" in Math::PlanePath),
(3Y+X)/2, (Y-X)/2 rotate -60
The first point N=1 is then along the X axis at X=2,Y=0. Or to have it diagonally upwards first then apply a mirroring -X before rotating
(3Y-X)/2, (Y+X)/2 mirror X and rotate -60
The plain -60 rotate puts the Nlevel=3^level point on the X axis for even number level, and at the top peak for odd level. With the extra mirroring it's the other way around. If drawing successive levels then the two ways can be alternated to have the endpoint on the X axis each time.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::SierpinskiArrowhead->new ()
$path = Math::PlanePath::SierpinskiArrowhead->new (align => $str)
Create and return a new arrowhead path object. align is a string, one of the following as described above.
"triangular" the default "right" "left" "diagonal"
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.
$n
$n < 0
If $n is not an integer then the return is on a straight line between the integer points.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, 3**$level).
(0, 3**$level)
The turn at N is given by ternary
turn(N) N + LowestNonZero(N) + CountLowZeros(N) ------- --------------------------------------- left even right odd
In the replications, turns N=1 and N=2 are both left. A low 0 digit expansion is mirror image to maintain initial segment direction. Parts "B" digit=1 above are each mirror images too so turns flip.
[flip for each 1 digit] [1 or 2] [flip for each low 0 digit]
N is odd or even according as the number of ternary 1 digits is odd or even (all 2 digits being even of course), so N parity accounts for the "B" mirrorings. On a binary computer this is just the low bit rather than examining the high digits of N. In any case if the ternary lowest non-0 is a 1 then it is not such a mirror so adding LowestNonZero cancels that.
This turn rule is noted by Alexis Monnerot-Dumaine in OEIS A156595. That sequence is LowestNonZero(N) + CountLowZeros(N) mod 2 and flipping according as N odd or even is the arrowhead turns.
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include,
http://oeis.org/A156595 (etc)
A156595 turn 0=left,1=right at even N=2,4,6,etc A189706 turn 0=left,1=right at odd N=1,3,5,etc A189707 (N+1)/2 of the odd N positions of left turns A189708 (N+1)/2 of the odd N positions of right turns align=diagonal A334483 X coordinate A334484 Y coordinate
Math::PlanePath, Math::PlanePath::SierpinskiArrowheadCentres, Math::PlanePath::SierpinskiTriangle, Math::PlanePath::KochCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
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