# SYNOPSIS

`````` use Math::PlanePath::QuadricIslands;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path is concentric islands made from four sides each an eight segment zig-zag (per the `QuadicCurve` path).

``````            27--26                     3
|   |
29--28  25  22--21             2
|       |   |   |
30--31  24--23  20--19         1
| 4--3          |
34--33--32    | 16--17--18     <- Y=0
|         1--2  |
35--36   7---8  15--14            -1
|   |       |
5---6   9  12--13            -2
|   |
10--11                -3

^
-3  -2  -1  X=0  1   2   3   4``````

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=5 through N=13 and the further sides to N=36. The individual sides are levels of the `QuadricCurve` path.

``````                                *---*
|   |
*---*     becomes     *---*   *   *---*
|   |
*---*
* <------ *
|         ^
|         |
|         |
v         |
* ------> *``````

The name `QuadricIslands` here is a slight mistake. Mandelbrot ("Fractal Geometry of Nature" 1982 page 50) calls any islands initiated from a square "quadric", not just this eight segment expansion. This curve also appears (unnamed) in Mandelbrot's "How Long is the Coast of Britain", 1967.

## Level Ranges

Counting the innermost square as level 0, each ring is

``````    length = 4 * 8^level     many points
Nlevel = 1 + length + ... + length[level-1]
= (4*8^level + 3)/7
Xstart = - 4^level / 2
Ystart = - 4^level / 2``````

For example the lower partial ring shown above is level 2 starting N=(4*8^2+3)/7=37 at X=-(4^2)/2=-8,Y=-8.

The innermost square N=1,2,3,4 is on 0.5 coordinates, for example N=1 at X=-0.5,Y=-0.5. This is centred on the origin and consistent with the (4^level)/2. Points from N=5 onwards are integer X,Y.

``````       4-------3    Y=+1/2
|       |
|   o   |
|
1-------2    Y=-1/2

X=-1/2   X=+1/2``````

# FUNCTIONS

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

`\$path = Math::PlanePath::QuadricIslands->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 * 8**\$level + 3) / 7,
(32 * 8**\$level - 4) / 7 )``````

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