Math::...rRule54(3pm)

💖 Math::...rRule54 — aka Math::PlanePath::CellularRule54 💖

okay bestie, sit down, hydrate, because this is a Perl module from the User Contributed Perl Documentation collection and it is so much cooler than it sounds. it's a plane path, it's a cellular automaton, it's giving math girlie excellence. no cap. 🦋

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✨ NAME ✨

Math::PlanePath::CellularRule54 — cellular automaton points. that's it, that's the whole vibe. 💅

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🌸 SYNOPSIS 🌸

the way you just use it:

use Math::PlanePath::CellularRule54;
my $path = Math::PlanePath::CellularRule54->new;
my ($x, $y) = $path->n_to_xy (123);

literally three lines and you're in. iconic. 🎀

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👑 DESCRIPTION 👑

so this is the pattern of Stephen Wolfram's "rule 54" cellular automaton

<http://mathworld.wolfram.com/Rule54.html>

arranged as rows. look at her:

   29 30 31  . 32 33 34  . 35 36 37  . 38 39 40     7
      25  .  .  . 26  .  .  . 27  .  .  . 28        6
         16 17 18  . 19 20 21  . 22 23 24           5
            13  .  .  . 14  .  .  . 15              4
                7  8  9  . 10 11 12                 3
                   5  .  .  .  6                    2
                      2  3  4                       1
                         1                      <- Y=0

   -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

the initial figure N=1,2,3,4 repeats in two-row groups with a 1 cell gap between figures. each two-row group has one extra figure, for a step of 4 more points than the previous two-row. growth era. 📈

the rightmost N on the even rows Y=0,2,4,6 etc is the hexagonal numbers N=1,6,15,28, etc — that's k*(2k-1). the hexagonal numbers of the "second kind" 1, 3, 10, 21, 36, etc — j*(2j+1) — are a steep sloping line upwards in the middle too. those two taken together are the triangular numbers 1,3,6,10,15 etc, k*(k+1)/2. she contains multitudes fr fr.

the 18-gonal numbers 18,51,100, etc are the vertical line at X=-3 on every fourth row Y=5,9,13, etc. iykyk. 💜

🌈 Row Ranges 🌈

the left end of each row is

Nleft = Y*(Y+2)/2 + 1     if Y even
        Y*(Y+1)/2 + 1     if Y odd

the right end is

Nright = (Y+1)*(Y+2)/2    if Y even
         (Y+1)*(Y+3)/2    if Y odd

       = Nleft(Y+1) - 1   ie. 1 before next Nleft

the row width Xmax-Xmin is 2*Y but with the gaps the number of visited points in a row is less than that, being either about 1/4 or 3/4 of the width on even or odd rows. she's selective, she's curating. 💅

rowpoints = Y/2 + 1        if Y even
            3*(Y+1)/2      if Y odd

for any Y of course the Nleft to Nright difference is the number of points in the row too

rowpoints = Nright - Nleft + 1

🧚 N Start 🧚

the default is to number points starting N=1 as shown above. an optional n_start can give a different start, in the same pattern. for example to start at 0,

n_start => 0

15 16 17    18 19 20    21 22 23           5
   12          13          14              4
       6  7  8     9 10 11                 3
          4           5                    2
             1  2  3                       1
                0                      <- Y=0

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

pick your own starting number, bestie, it's your world. ⭐

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🫶 FUNCTIONS 🫶

see "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. that's the family lore.

• $path = Math::PlanePath::CellularRule54->new ()
• $path = Math::PlanePath::CellularRule54->new (n_start => $n)

create and return a new path object. baby's first object. 🍼

• ($x,$y) = $path->n_to_xy ($n)

return the X,Y coordinates of point number $n on the path.

• $n = $path->xy_to_n ($x,$y)

return the point number for coordinates "$x,$y". $x and $y are each rounded to the nearest integer, which has the effect of treating each cell as a square of side 1. if "$x,$y" is outside the pyramid or on a skipped cell the return is "undef". 💀 so check your returns, bestie, don't get caught lacking.

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🔥 OEIS 🔥

this pattern is in Sloane's Online Encyclopedia of Integer Sequences in a couple of forms,

<http://oeis.org/A118108> (etc)

A118108    whole-row used cells as bits of a bignum
A118109    1/0 used and unused cells across rows

two entries, both bangers, both cited. slay. ✨

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🌟 SEE ALSO 🌟

• Math::PlanePath, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule57, Math::PlanePath::CellularRule190, Math::PlanePath::PyramidRows
• Cellular::Automata::Wolfram
• <http://mathworld.wolfram.com/Rule54.html>

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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/>.

bestie this part is the serious part, so listen: no warranty, like, at all. it's free software under the GNU General Public License version 3 or later, and you gotta keep the license along with it. read the whole thing if you're gonna redistribute, okay? protecting the vibes and the legalese. 🫶

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okay that's the whole page!! thanks for reading my little hyperfixation. this one's from perl v5.32.0, dated 2021-01-23, and it lives in the User Contributed Perl Documentation. mwah, go draw some cellular automata, you main character. 💖✨🦋🎀💅