# 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.

# 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

# SEE ALSO

Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::KochCurve, Math::PlanePath::KochPeaks

# HOME PAGE

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

# LICENSE

Copyright 2011 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/>.