Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::SierpinskiArrowhead -- self-similar triangular path traversal
SYNOPSIS
use Math::PlanePath::SierpinskiArrowhead;
my $path = Math::PlanePath::SierpinskiArrowhead->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
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
* * * * * * * * * * * *
* * * * * * * * * * *
* * * * * * * *
* * * * *
* * * * * * *
* * * * *
* * * *
* * *
* * * * * * * * * * *
* * * * * * * * * * *
* * * * * * * *
* * * * *
* * * * * * *
* * * * *
* * * *
* * *
* * * * * *
* * * * *
* * * *
* * *
* * *
* * *
* *
*
*
Sierpinski Triangle
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.
The attraction of this Arrowhead path is that it makes a connected traversal through the Sierpinski
triangle pattern.
Level Sizes
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)
Align Parameter
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.
"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
Sideways
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.
FUNCTIONS
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.
If $n is not an integer then the return is on a straight line between the integer points.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 3**$level)".
FORMULAS
Turn
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.
OEIS
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
SEE ALSO
Math::PlanePath, Math::PlanePath::SierpinskiArrowheadCentres, Math::PlanePath::SierpinskiTriangle,
Math::PlanePath::KochCurve
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU
General Public License as published by the Free Software Foundation; either version 3, or (at your
option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
License for more details.
You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see
<http://www.gnu.org/licenses/>.
perl v5.32.0 2021-01-23 Math::PlanePath...pinskiArrowhead(3pm)