NAME
Math::PlanePath::KochSnowflakes -- Koch snowflakes as concentric rings
SYNOPSIS
use Math::PlanePath::KochSnowflakes;
my $path = Math::PlanePath::KochSnowflakes->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
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 9The 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.
Triangular Coordinates
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=+2/3
*
/ \
/ \
/ \
/ o \
/ \
N=1 *-----------* N=2
X=-1, Y=-1/3 X=1, Y=-1/3These 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.
Level Ranges
Counting the innermost triangle as level 0, each ring is
Nstart = 4^level
length = 3*4^level many pointsFor 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^levelNotice 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 for a complete program doing that as an SVG image file.
Area
The area of the snowflake at a given level can be calculated from the area under the Koch curve per "Area" in Math::PlanePath::KochCurve which is the 3 sides, and the central triangle
* ^ Yhi
/ \ | height = 3^level
/ \ |
/ \ |
*-------* v
<-------> width = 3^level - (- 3^level) = 2*3^level
Xlo Xhi
triangle_area = width*height/2 = 9^level
snowflake_area[level] = triangle_area[level] + 3*curve_area[level]
= 9^level + 3*(9^level - 4^level)/5
= (8*9^level - 3*4^level) / 5If the snowflake is conceived as a fractal of fixed initial triangle size and ever-smaller notches then the area is divided by that central triangle area 9^level,
unit_snowflake[level] = snowflake_area[level] / 9^level
= (8 - 3*(4/9)^level) / 5
-> 8/5 as level -> infinityWhich is the well-known 8/5 * initial triangle area for the fractal snowflake.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
Level Methods
($n_lo, $n_hi) = $path->level_to_n_range($level)-
Return per "Level Ranges" above,
(4**$level, 4**($level+1) - 1)
FORMULAS
Rectangle to N Range
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) - 1Using 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
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to the Koch snowflake include the following. See "OEIS" in Math::PlanePath::KochCurve for entries related to a single Koch side.
http://oeis.org/A164346 (etc)
A164346 number of points in ring n, being 3*4^n
A178789 number of acute angles in ring n, 4^n + 2
A002446 number of obtuse angles in ring n, 2*4^n - 2The acute angles are those of +/-120 degrees and the obtuse ones +/-240 degrees. Eg. in the outer ring=2 shown above the acute angles are at N=18, 22, 24, 26, etc. The angles are all either acute or obtuse, so A178789 + A002446 = A164346.
SEE ALSO
Math::PlanePath, Math::PlanePath::KochCurve, Math::PlanePath::KochPeaks
Math::PlanePath::QuadricIslands
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/>.