NAME
Math::PlanePath::SierpinskiArrowheadCentres -- self-similar triangular path traversal
SYNOPSIS
use Math::PlanePath::SierpinskiArrowheadCentres;
my $path = Math::PlanePath::SierpinskiArrowheadCentres->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This is a version of the Sierpinski arrowhead path taking the centres of each triangle represented by the arrowhead segments. The effect is to traverse the Sierpinski triangle.
... ...
/ /
. 30 . . . . . 65 . ...
/ \ /
28----29 . . . . . . 66 68 9
\ \ /
27 . . . . . . . 67 8
\
26----25 19----18----17 15----14----13 7
/ \ \ / /
24 . 20 . 16 . 12 6
\ / /
23 21 . . 10----11 5
\ / \
22 . . . 9 4
/
4---- 5---- 6 8 3
\ \ /
3 . 7 2
\
2---- 1 1
/
0 <- Y=0
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7The base figure is the N=0 to N=2 shape. It's repeated up in mirror image as N=3 to N=6 then across rotated as N=6 to N=9. At the next level the same is done with the N=0 to N=8 shape, up mirrored as N=9 to N=17 and across rotated as N=18 to N=26, etc.
The X,Y coordinates are on a triangular lattice using every second integer X, per "Triangular Lattice" in Math::PlanePath.
The base pattern is a triangle like
.-------.-------.
\ / \ /
\ 2 / m \ 1 /
\ / \ /
.- - - -.
\ /
\ 0 /
\ /
.Higher levels replicate this within the triangles 0,1,2 but the middle "m" is not traversed. The result is the familiar Sierpinski triangle by connected steps 2 across or 1 diagonal.
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * *
* * * *
* *
* * * * * * * *
* * * *
* * * *
* *
* * * *
* *
* *
*See the SierpinskiTriangle path to traverse by rows instead.
Level Ranges
Counting the N=0,1,2 part as level 1, each replication level goes from
Nstart = 0
Nlevel = 3^level - 1 inclusiveFor example level 2 from N=0 to N=3^2-1=9. Each level doubles in size,
0 <= Y <= 2^level - 1
- (2^level - 1) <= X <= 2^level - 1The Nlevel position is alternately on the right or left,
Xlevel = / 2^level - 1 if level even
\ - 2^level + 1 if level oddThe Y axis ie. X=0, is crossed just after N=1,5,17,etc which is is 2/3 through the level, which is after two replications of the previous level,
Ncross = 2/3 * 3^level - 1
= 2 * 3^(level-1) - 1FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::SierpinskiArrowheadCentres->new ()-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.If
$nis not an integer then the return is on a straight line between the integer points.
FORMULAS
Rectangle to N Range
An easy over-estimate of the range can be had from inverting the Nlevel formulas in "Level Ranges" above.
level = floor(log2(Ymax)) + 1
Nmax = 3^level - 1For example Y=5, level=floor(log2(11))+1=3, so Nmax=3^3-1=26, which is the left end of the Y=7 row, ie. rounded up to the end of the Y=4 to Y=7 replication.
SEE ALSO
Math::PlanePath, Math::PlanePath::SierpinskiArrowhead, Math::PlanePath::SierpinskiTriangle
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2011, 2012 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/>.