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 zigzag (per the QuadicCurve
path).
2726 3
 
2928 25 2221 2
   
3031 2423 2019 1
 43 
343332  161718 < Y=0
 12 
3536 78 1514 1
  
56 9 1213 2
 
1011 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[level1]
= (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.
43 Y=+1/2
 
 o 

12 Y=1/2
X=1/2 X=+1/2
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
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/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.