NAME

Math::PlanePath::MathImageKochSquareflakes -- four-sided Koch snowflakes

SYNOPSIS

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

DESCRIPTION

In progress.

This is the Koch curve pattern arranged as concentric four-sided snowflakes.

                              61                                10
                             /  \
                        63-62    60-59                           9
                         |           |
               67       64          58       55                  8
              /  \     /              \     /  \
         69...    66-65                57-56    54-53            7
          |                                         |
         70                                        52            6
        /                                            \
      71                                              51         5
        \                                            /
         72                                        50            4
          |                                         |
         73                   15                   49            3
        /                    /  \                    \
   75-74                17-16    14-13                48-47      2
    |                    |           |                    |
   76                   18          12                   46      1
  /                    /     4---3    \                    \
77                   19        . |     11                   45  Y=0
  \                    \     1---2    /                    /
   78                   20          10                   44     -1
    |                                |                    |
   79-80                 5--6     8--9                42-43     -2
        \                    \  /                    /
         81                    7                   41           -3
          |                                         |
         82                                        40           -4
        /                                            \
      83                                              39        -5
        \                                            /
         84                                        38           -6
                                                    |
         21-22    24-25                33-34    36-37           -7
              \  /     \              /     \  /
               23       26          32       35                 -8
                         |           |
                        27-28    30-31                          -9
                             \  /
                              29                               -10

                               ^
   -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9 10

The innermost square N=1 to N=4 is the initial figure. Its sides expand in the Koch curve pattern in subsequent rings. The initial figure is on X=+/-0.5,Y=+/-0.5 fractions but the points after that are integer X,Y.

Inward

The inward option can direct the sides inward instead of outward. The shape and side lengths etc are the same.

69-68    66-65                57-56    54-53     7
 |   \  /     \              /     \  /    |
70    67       64          58       55    52     6
  \             |           |            /
   71          63-62    60-59          51        5
  /                 \  /                 \
72                   61                   50     4
 |                                         |
73                                        49     3
  \                                      /
   74-75       17-16    14-13       47-48        2
       |        |   \  /    |        |
      76       18    15  3 12       46           1
        \        \  4--3  /        /
         77       19   |11       45          <- Y=0
        /        /  1--2  \        \
      78       20     7    10       44          -1
       |            /  \    |        |
   80-79        5--6     8--9       43-42       -2
  /                                      \
81                                        41    -3
 |                                         |
82                   29                   40    -4
  \                 /  \                 /
   83          27-28    30-31          39       -5
  /             |           |            \
84    23       26          32       35    38    -6
     /  \     /              \     /  \    |
21-22    24-25                33-34    36-37    -7

                      ^
-7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

Level Ranges

Counting the innermost N=1 to N=4 square as level 0, a given level has

looplen = 4*4^level

many points. The start of a level is therefore N=1 plus preceding loop lengths,

Nstart = 1 + 4*[ 1 + 4 + 4^2 + ... + 4^(level-1) ]
       = 1 + 4*(4^level - 1)/3
       = (4^(level+1) - 1)/3

and the end similarly as the total loop lengths inclusive, and also simply one less than the next Nstart,

Nend = 4 * [ 1 + ... + 4^level ]
     = (4^(level+2) - 4) / 3

     = Nstart(level+1) - 1

For example,

level  Nstart   Nend
  0       1       4
  1       5      20
  2      21      84
  3      85     340

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

$path = Math::PlanePath::MathImageKochSquareflakes->new ()

$path = Math::PlanePath::MathImageKochSquareflakes->new (inward => $bool)

Create and return a new path object.

SEE ALSO

Math::PlanePath, Math::PlanePath::KochCurve, Math::PlanePath::KochSnowflakes