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 is an integer version of the Sierpinski arrowhead path. It follows a
self-similar triangular shape leaving middle triangle gaps.
\
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¶
The arrowhead is sometimes drawn on its side, with a base along the X axis. That
can be had 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 could 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)".
OEIS¶
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include,
A189706 turn 0=left,1=right at odd positions 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
A156595 turn 0=left,1=right at even positions N=2,4,6,etc
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 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/>.