and 1 contributors

# 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 for the Y coordinates of the innermost triangle which is

``````                    N=3
X=0, Y=+0.666
/                \
N=1                     N=2
X=-1, Y=-0.333  ------   X=1, Y=-0.333``````

These values are consistent with the centring and scaling of the higher levels. Rounding to an integer gives Y=0 or Y=1 and doesn't overlap the subsequent points if all-integer is desired.

## 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 = -(2/3)*3^level
Yhi = +(2/3)*3^level``````

except for the initial triangle which doesn't have a downward notch and is only Y=-1/3 not Y=-2/3.

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. At level 1 they equal the corners horizontally, ie. N=14 is at X=-3 the same as N=4, and on the right N=10 at X=+3 the same as N=8, but no more than that.

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 for a 2-wide 1-high figure of desired fineness. See examples/koch-svg.pl for an complete program doing that as an SVG image file.

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the 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^level) <= Y <= 2*(3^level)``````

So the maximum X,Y in a rectangle gives a level,

``    level = ceil(log3(max(x1, x2, y1/2, y2/2)))``

and the last point in that level is

``     N = 4^(level+1) - 1``

Using that as an N range is an over-estimate, but an easy calculation. It's not too difficult to trace down for an exact range