Math::...exPlus(3pm)

🌸 NAME 🌸

Math::PlanePath::ComplexPlus — points in complex base i+r. That's the whole vibe, bestie. Also heads up: this page is filed under the User Contributed Perl Documentation manual, so that's where you are rn. 💅

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✨ SYNOPSIS ✨

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

Look at that. Three lines. Iconic. 🫶

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🦋 DESCRIPTION 🦋

So this path traverses points by a complex number base i+r for integer r>=1. The default? Base i+1. Periodt.

                            30  31          14  15                 5
                        28  29          12  13                     4
                            26  27  22  23  10  11   6   7         3
                        24  25  20  21   8   9   4   5             2
            62  63          46  47  18  19           2   3         1
        60  61          44  45  16  17           0   1         <- Y=0
            58  59  54  55  42  43  38  39                        -1
        56  57  52  53  40  41  36  37                            -2
                    50  51  94  95  34  35  78  79                -3
                48  49  92  93  32  33  76  77                    -4
                            90  91  86  87  74  75  70  71        -5
                        88  89  84  85  72  73  68  69            -6
           126 127         110 111  82  83          66  67        -7
       124 125         108 109  80  81          64  65            -8
           122 123 118 119 106 107 102 103                        -9
       120 121 116 117 104 105 100 101                           -10
                   114 115          98  99                       -11
               112 113          96  97                           -12

         ^   ^   ^   ^   ^   ^   ^   ^   ^   ^   ^   ^   ^
       -10  -9  -8  -7  -6  -5  -4  -3  -2  -1  X=0  1   2

The shape of these points N=0 to N=2^k-1 inclusive is equivalent to the twindragon turned 135 degrees. Each complex base point corresponds to a unit square inside the twindragon curve (two DragonCurve back-to-back). Two dragons, one square each, no cap. 🐉🐉

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🌈 Real Part 🌈

Option realpart => $r selects another r for complex base b=i+r. For example:

realpart => 2
                                        45 46 47 48 49      8
                                  40 41 42 43 44            7
                            35 36 37 38 39                  6
                      30 31 32 33 34                        5
                25 26 27 28 29 20 21 22 23 24               4
                         15 16 17 18 19                     3
                   10 11 12 13 14                           2
              5  6  7  8  9                                 1
        0  1  2  3  4                                   <- Y=0

        ^
       X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15

N is broken into digits of a base norm=r*r+1, ie. digits 0 to r*r inclusive. Get it:

norm = r*r + 1
Nstart = 0
Nlevel = norm^level - 1

The low digit of N makes horizontal runs of r*r+1 many points, such as N=0 to N=4, then N=5 to N=9, etc shown above. In the default r=1 these runs are 2 long. For r=2 shown above they're 2*2+1=5 long, or r=3 would be 3*3+1=10, etc. Math is mathing. 🔥

The offset for each successive run is i+r, ie. Y=1,X=r such as at N=5 shown above. Then the offset for the next level is (i+r)^2 = (2r*i + r^2-1) so N=25 begins at Y=2*r=4, X=r*r-1=3. In general each level adds an angle:

angle = atan(1/r)
Nlevel_angle = level * angle

So the points spiral around anti-clockwise. For r=1 the angle is atan(1/1)=45 degrees, so that for example level=4 is angle 4*45=180 degrees, putting N=2^4=16 on the negative X axis as shown in the first sample above. Spinning, spiraling, main character energy. 🌀

As r becomes bigger the angle becomes smaller, making it spiral more slowly. The points never fill the plane, but the set of points N=0 to Nlevel are all touching. They're all up in each other's business. 🫶

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🎀 Arms 🎀

For realpart => 1, an optional arms => 2 adds a second copy of the curve rotated 180 degrees and starting from X=0,Y=1. It meshes perfectly to fill the plane. Each arm advances successively so N=0,2,4,etc is the plain path and N=1,3,5,7,etc is the copy. Slay, they interlock. 💞

arms=>2

    60  62          28  30                                 5
56  58          24  26                                     4
    52  54  44  46  20  22  12  14                         3
48  50  40  42  16  18   8  10                             2
                36  38   3   1   4   6  35  33                 1
            32  34   7   5   0   2  39  37                 <- Y=0
                        11   9  19  17  43  41  51  49        -1
                    15  13  23  21  47  45  55  53            -2
                                27  25          59  57        -3
                            31  29          63  61            -4

                             ^
-6  -5  -4  -3  -2  -1  X=0  1   2   3   4   5   6

There's no arms parameter for other realpart values as yet, only for i+1. Is there a good rotated arrangement for others? Do norm many copies fill the plane in general? Asking for a friend (the friend is us, we don't know either, the manpage is literally asking too 😭).

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💜 FUNCTIONS 💜

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. All the shared stuff lives there, go look.

• $path = Math::PlanePath::ComplexPlus->new () — create and return a new path object. Fresh path, fresh me. ✨
• ($x,$y) = $path->n_to_xy ($n) — return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list. Negative N? Nothing. Void. Ghosted. 👻

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🌟 Level Methods 🌟

• ($n_lo, $n_hi) = $path->level_to_n_range($level) — return (0, 2**$level - 1), or for 2 arms return (0, 2 * 2**$level - 1). With the realpart option return (0, $norm**$level - 1) where norm=realpart^2+1.

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📐 FORMULAS 📐

Various formulas and pictures etc for the i+1 case can be found in the author's long mathematical write-up (section "Complex Base i+1"):

<http://user42.tuxfamily.org/dragon/index.html>

Long mathematical write-up = the author really said "let me cook". 👑

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🔢 OEIS 🔢

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include:

<http://oeis.org/A290885> (etc)

realpart=1 (i+1, the default)
  A290885    -X
  A290884    Y
  A290886    norm X^2 + Y^2
  A146559    dX at N=2^k-1 (step to next replication level)
  A077950,A077870
           location of ComplexMinus origin in ComplexPlus
           (mirror horizontal even level, vertical odd level)

Number sequences for the girlies who love a lookup table. 🔍

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👀 SEE ALSO 👀

Math::PlanePath, Math::PlanePath::ComplexMinus, Math::PlanePath::ComplexRevolving, Math::PlanePath::DragonCurve

<http://user42.tuxfamily.org/dragon/index.html>

Sibling paths, all valid, all iconic. 🌈

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🏠 HOME PAGE 🏠

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

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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/>.

fr fr read the license, bestie, it's giving legal but make it cute. No warranty, no promises, but all the freedom. 💖

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and that's the whole page!! ✨ this one's from perl v5.32.0, dated 2021-01-23, and it's the Math::...exPlus(3pm) page — go build yourself a twindragon, you've earned it. bye bestie 💅🦋🎀