Math::PlanePath::QuadricIslands -- quadric curve rings
use Math::PlanePath::QuadricIslands; my $path = Math::PlanePath::QuadricIslands->new; my ($x, $y) = $path->n_to_xy (123);
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.
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
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::QuadricIslands->new ()
Create and return a new path object.
($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 )
Math::PlanePath, Math::PlanePath::QuadricCurve, Math::PlanePath::KochSnowflakes, Math::PlanePath::GosperIslands
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
This file is part of Math-PlanePath.
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