Kevin Ryde > Math-PlanePath-114 > Math::PlanePath::KochSnowflakes

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Module Version: 114   Source   Latest Release: Math-PlanePath-115

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 in the Math-PlanePath sources 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.

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.

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

HOME PAGE ^

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE ^

Copyright 2011, 2012, 2013 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/>.

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