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Math::PlanePath::KochSnowflakes -- Koch snowflakes as concentric rings
Contents
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 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.
TriangularCoordinates
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=-1/3
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.
LevelRanges
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 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) / 5
If 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 -> infinity
Which is the well-known 8/5 * initial triangle area for the fractal snowflake.
Formulas
RectangletoNRange
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
Functions
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::KochSnowflakes->new ()"
Create and return a new path object.
LevelMethods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return per "Level Ranges" above,
(4**$level,
4**($level+1) - 1)
Home Page
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::KochSnowflakes(3pm)
Name
Math::PlanePath::KochSnowflakes -- Koch snowflakes as concentric rings
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 - 2
The 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
Synopsis
use Math::PlanePath::KochSnowflakes;
my $path = Math::PlanePath::KochSnowflakes->new;
my ($x, $y) = $path->n_to_xy (123);
PNG Icons
Emojis