Math::...exArms(3pm)

🌸 Math::PlanePath::HexArms 🌸

okay bestie, so this is the manual page Math::...exArms(3pm) (the terminal just truncates the middle, it's really Math::PlanePath::HexArms(3pm) 💅) and it's literally about six spiral arms and I'm obsessed 🌀✨

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✨ NAME ✨

Math::PlanePath::HexArms — six spiral arms 🦋 (yes that's the whole thing, short and iconic)

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🎀 SYNOPSIS 🎀

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

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💖 DESCRIPTION 💖

So this path follows six spiral arms, and each one advances successively 💫 like a little conga line of integers, no cap:

                                          ...--66                      5
                                                 \
                    67----61----55----49----43    60                   4
                   /                         \      \
                ...    38----32----26----20    37    54                3
                      /                    \     \     \
                    44    21----15---- 9    14    31    48   ...       2
                   /     /              \      \    \     \     \
                 50    27    10---- 4     3     8    25    42    65    1
                 /    /     /                 /     /     /     /
              56    33    16     5     1     2    19    36    59    <-Y=0
             /     /     /     /        \        /     /     /
           62    39    22    11     6     7----13    30    53         -1
             \     \     \     \     \              /     /
             ...    45    28    17    12----18----24    47            -2
                      \     \     \                    /
                       51    34    23----29----35----41   ...         -3
                         \     \                          /
                          57    40----46----52----58----64            -4
                            \
                             63--...                                  -5

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

The X,Y points are integers and they use every second position to give a triangular lattice 🧚 per "Triangular Lattice" in Math::PlanePath.

Each arm is N=6*k+rem for a remainder rem=0,1,2,3,4,5, so bestie, if you've got sequences related to multiples of 6 or with a modulo 6 pattern, they might land on particular arms 👀✨

🍓 Abundant Numbers 🍓

The "abundant" numbers are those N where the sum of proper divisors is > N. For example, 12 is abundant because it's divisible by 1,2,3,4,6 and their sum is 16. And literally all multiples of 6 starting from 12 are abundant 💀. Plotting the abundant numbers on the path gives the 6*k arm and some other points in between, like:

                   * * * * * * * * * * * *   *   *   ...
                  *                       *           *
                 *   *   *           *     *   *       *
                *                           *           *
               *           *                 *           *
              *                           *   *           *
             *           * * * * * *           *       *   *
            *           *           *   *       *           *
           *   *   *   *         *   *           *       *   *
          *           *               *   *   *   *           *
         *   *   *   *                 *           *   *       *
        *           *   *             *   *       *           *
       *       *   *                 *           *           *
        *           *           * * *           *           *
         *           *                 *       *           *
          *   *       *   *   *           *   *           *
           *           *                     *   *       *
            *           *       *           *           *
             *   *       *                 *   *   *   *
              *           * * * * * * * * *           *
               *   *                         *       *
                *         *       *                 *
                 *   *                         *   *
                  *         *       *       *     *
                   *                             *
                    * * * * * * * * * * * * * * *

There's blank arms either side of the 6*k because 6*k+1 and 6*k-1 are not abundant until some fairly big values 😭. The first abundant 6*k+1 might be 5,391,411,025, and the first 6*k-1 might be 26,957,055,125. like, big big. huge even.

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👑 FUNCTIONS 👑

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes 💜

• $path = Math::PlanePath::HexArms->new () — Create and return a new square spiral object. (yes bestie it says "square spiral object", we don't ask questions, we just vibe 💅)
• ($x,$y) = $path->n_to_xy ($n) — Return the X,Y coordinates of point number $n on the path ✨

For $n < 1 the return is an empty list, as the path starts at 1 🫶

Fractional $n gives a point on the line between $n and $n+6, that $n+6 being the next on the same spiralling arm 🌀. This is probably of limited use, but arises fairly naturally from the calculation. (ngl kinda niche but okay, it's giving honest documentation 😭)

🌈 Descriptive Methods 🌈

• $arms = $path->arms_count() — Return 6. periodt. 💅✨

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💜 SEE ALSO 💜

Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondArms, Math::PlanePath::HexSpiral — go read the whole family bestie, they're 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. 🚨 (that's the serious bit, don't skip it, fr fr)

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.

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and that's the whole page bestie 🎀 this was Math::...exArms(3pm) from perl v5.32.0, dated 2021-01-23 — go forth and spiral, you're the main character now 👑💖✨