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math-image-66
River stage zero No dependents
/ Math::PlanePath::MathImageSquareflakes

NAME

Math::PlanePath::MathImageSquareflakes -- Koch snowflakes as concentric rings

SYNOPSIS

use Math::PlanePath::MathImageSquareflakes;
my $path = Math::PlanePath::MathImageSquareflakes->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path traces out concentric integer versions of the Koch snowflake at successively greater iteration levels.

                            ^
-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 square N=1,2,3,4 then for the next level each straight side expands to 4x longer and a zigzag like N=4 through N=12,

                              *---*
                              |   |
*---*     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.

Level Ranges

Counting the innermost triangle as level 0, each ring is

Nstart = 8^level
length = 4*(8^level)   many points

For example the outer ring shown above is level 2 starting N=8^2=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

$path = Math::PlanePath::MathImageSquareflakes->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

SEE ALSO

Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::KochSnowflakes