---
name: "Math::PlanePath::QuadricIslands"
section: "3pm"
platform: "linux"
package: "libmath-planepath-perl"
version: "129-1"
release: "Debian 13 (Trixie)"
platforms: ["linux"]
url: "https://new.carta.tech/Math::PlanePath::QuadricIslands"
---

# Math::PlanePath::QuadricIslands(3pm)

## NAME

Math::PlanePath::QuadricIslands -- quadric curve rings

## SYNOPSIS

```
 use Math::PlanePath::QuadricIslands;
 my $path = Math::PlanePath::QuadricIslands->new;
 my ($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[0] + ... + 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 )
  ```

## SEE ALSO

Math::PlanePath, Math::PlanePath::QuadricCurve, Math::PlanePath::KochSnowflakes, Math::PlanePath::GosperIslands

## HOME PAGE

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

## LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

This file is part of Math-PlanePath.

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/\>.