 Math::PlanePath::KochSnowflakes(3) 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.

## Level Methods

"(\$n_lo, \$n_hi) = \$path->level_to_n_range(\$level)"
Return per ``Level Ranges'' above,

```    (4**\$level,
4**(\$level+1) - 1)
```

## 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.