Math::...Islands(3pm)

💖 Math::...Islands 💖

okay bestie, sit DOWN, because we're talking about Math::PlanePath::QuadricIslands — quadric curve rings, and honestly? it's giving concentric island realness 🏝️✨

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🌸 NAME 🌸

Math::PlanePath::QuadricIslands — quadric curve rings 💅

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✨ SYNOPSIS ✨

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

that's it. that's the whole vibe. you make a path, you ask it for coordinates, slay 🎀

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🦋 DESCRIPTION 🦋

so this path is literally concentric islands — each one built from four sides, and every side is an eight segment zig-zag (per the "QuadicCurve" path). iykyk 🌈

                   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, and then for the next level every straight side expands to 4x longer and goes zigzag like N=5 through N=13, with the further sides going all the way to N=36. the individual sides? yeah, they're levels of the "QuadricCurve" path. no cap 🫶

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

now bestie, tea time ☕ — 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. and this curve also shows up (unnamed, rude honestly) in Mandelbrot's "How Long is the Coast of Britain", 1967. 💀

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👑 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. math mathing fr fr 🔥

the innermost square N=1,2,3,4 sits on 0.5 coordinates — for example N=1 at X=-0.5,Y=-0.5. it's centred on the origin and consistent with the (4^level)/2. points from N=5 onwards are integer X,Y. periodt 💅

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

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

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🎀 FUNCTIONS 🎀

see "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. the shared lore, if you will 🌸

• $path = Math::PlanePath::QuadricIslands->new () — create and return a new path object. your main character moment starts here ✨

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🧚 Level Methods 🧚

• ($n_lo, $n_hi) = $path->level_to_n_range($level) — return per "Level Ranges" above, which is:

    ( ( 4 * 8**$level + 3) / 7,
      (32 * 8**$level - 4) / 7 )

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🌈 SEE ALSO 🌈

• Math::PlanePath
• Math::PlanePath::QuadricCurve
• Math::PlanePath::KochSnowflakes
• Math::PlanePath::GosperIslands

the whole friend group, obsessed 🫶

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💜 HOME PAGE 💜

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

go click it bestie, it's iconic 🔥

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⭐ 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/>. 💖

ok one lil disclaimer bestie — this is perl v5.32.0, dated 2021-01-23, and it's User Contributed Perl Documentation for the Math::...Islands(3pm) page. take care of her, she's a queen 👑💅✨