Math::PlanePath::KochSnowflakes -- Koch snowflakes as concentric rings
use Math::PlanePath::KochSnowflakes; my $path = Math::PlanePath::KochSnowflakes->new; my ($x, $y) = $path->n_to_xy (123);
This path traces out concentric integer versions of the Koch snowflake at successively greater iteration levels.
48 6 / \ 50----49 47----46 5 \ / 54 51 45 42 4 / \ / \ / \ 56----55 53----52 44----43 41----40 3 \ / 57 12 39 2 / / \ \ 58----59 14----13 11----10 37----38 1 \ \ 3 / / 60 15 1----2 9 36 <- Y=0 / \ \ 62----61 4---- 5 7---- 8 35----34 -1 \ \ / / 63 6 33 -2 \ 16----17 19----20 28----29 31----32 -3 \ / \ / \ / 18 21 27 30 -4 / \ 22----23 25----26 -5 \ / 24 -6 ^ -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
The initial figure is the triangle N=1,2,3 then for the next level each straight side expands to 3x longer and a notch like N=4 through N=8,
*---* becomes *---* *---* \ / *
The angle is maintained in each replacement, for example the segment N=5 to N=6 becomes N=20 to N=24 at the next level.
The X,Y coordinates are arranged as integers on a square grid per "Triangular Lattice" in Math::PlanePath, except the Y coordinates of the innermost triangle which is
N=3 X=0, Y=+0.666 / \ / \ / \ / \ N=1 N=2 X=-1, Y=-0.333 ------ X=1, Y=-0.333
These values are not integers, but they're consistent with the centring and scaling of the higher levels. If all-integer is desired then rounding gives Y=0 or Y=1 and doesn't overlap the subsequent points.
Counting the innermost triangle as level 0, each ring is
Nstart = 4^level length = 3*(4^level) many points
For example the outer ring shown above is level 2 starting N=4^2=16 and having length=3*4^2=48 points (through to N=63 inclusive).
The X range at a given level is the initial triangle baseline iterated out. Each level expands the sides by a factor of 3 so
Xlo = -(3^level) Xhi = +(3^level)
For example level 2 above runs from X=-9 to X=+9. The Y range is the points N=6 and N=12 iterated out. Ylo in level 0 since there's no downward notch on that innermost triangle.
Ylo = / -(2/3)*3^level if level >= 1 \ -1/3 if level == 0 Yhi = +(2/3)*3^level
Notice that for each level the extents grow by a factor of 3 but the notch introduced in each segment is not big enough to go past the corner positions. They can equal the extents horizontally, for example in level 1 N=14 is at X=-3 the same as the corner N=4, and on the right N=10 at X=+3 the same as N=8, but they don't go past.
The snowflake is an example of a fractal curve with ever finer structure. The code here can be used for that by going from N=Nstart to N=Nstart+length-1 and scaling X/3^level Y/3^level to give a 2-wide 1-high figure of desired fineness. See examples/koch-svg.pl in the Math-PlanePath sources for a complete program doing that as an SVG image file.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::KochSnowflakes->new ()
Create and return a new path object.
As noted in "Level Ranges" above, for a given level
-(3^level) <= X <= 3^level -(2/3)*(3^level) <= Y <= (2/3)*(3^level)
So the maximum X,Y in a rectangle gives
level = ceil(log3(max(abs(x1), abs(x2), abs(y1)*3/2, abs(y2)*3/2)))
and the last point in that level is
Nlevel = 4^(level+1) - 1
Using this as an N range is an over-estimate, but an easy calculation. It's not too difficult to trace down for an exact range
Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::KochCurve, Math::PlanePath::KochPeaks
http://user42.tuxfamily.org/math-planepath/index.html
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/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.