--- name: "Math::PlanePath::KochSnowflakes" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::KochSnowflakes" --- # Math::PlanePath::KochSnowflakes(3pm) ## 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 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. ### 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=-1/3 ``` 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. ### Level Ranges 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. ## 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. ### 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) - 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 ## 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. \ (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 ## 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 \.