Math::...Groups(3pm)

💖 Math::...Groups — the Math::PlanePath::DigitGroups era 💖

okay bestie, listen. this is Math::PlanePath::DigitGroups, and it lives in the User Contributed Perl Documentation section of your man pages (yes, 3pm, the cute one). it's a plane path class, it splits digits into groups, and it's genuinely iconic. let's go. 🦋

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✨ NAME ✨

Math::PlanePath::DigitGroups — X,Y digits grouped by zeros

like. it takes a number N, and splits it into an X and a Y by grouping the digits between the zeros. that's the whole vibe. 💅

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🌸 SYNOPSIS 🌸

use Math::PlanePath::DigitGroups;

my $path = Math::PlanePath::DigitGroups->new (radix => 2);
my ($x, $y) = $path->n_to_xy (123);

three lines. that's it. that's the whole entrance. slay. ✨

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👑 DESCRIPTION 👑

so what this path does is it splits an N into X and Y by digit groups separated by a 0. the default is binary, so like, watch this:

N = 110111001011

gets split into groups with a leading high 0 bit, and then those groups go to X and Y alternately, like a little relay race:

N = 11 0111 0 01 011
     X   Y  X  Y  X

X = 11      0    011 = 110011
Y =    0111   01     =  11101

the result? a one-to-one mapping between numbers N>=0 and pairs X>=0,Y>=0. no collisions, no leftovers, every N gets exactly one home. it's giving perfect bijection energy. 🫶

the default binary is:

11  |   38   77   86  155  166  173  182  311  550  333  342  347
10  |   72  145  148  291  168  297  300  583  328  337  340  595
 9  |   66  133  138  267  162  277  282  535  322  325  330  555
 8  |  128  257  260  515  272  521  524 1031  320  545  548 1043
 7  |   14   29   46   59  142   93  110  119  526  285  302  187
 6  |   24   49   52   99   88  105  108  199  280  177  180  211
 5  |   18   37   42   75   82   85   90  151  274  165  170  171
 4  |   32   65   68  131   80  137  140  263  160  161  164  275
 3  |    6   13   22   27   70   45   54   55  262  141  150   91
 2  |    8   17   20   35   40   41   44   71  136   81   84   83
 1  |    2    5   10   11   34   21   26   23  130   69   74   43
Y=0 |    0    1    4    3   16    9   12    7   64   33   36   19
    +-------------------------------------------------------------
       X=0    1    2    3    4    5    6    7    8    9   10   11

okay so the lore behind the X axis: N=0,1,4,3,16,9,etc along the X axis is X with zero bits doubled. for example X=9 is binary 1001, so you double up the zero bits to 100001 for N=33 at X=9,Y=0. why? because in the digit groups Y=0, so when X is grouped by its zero bits there's an extra 0 from Y sitting in between each group. iykyk. 😌

and same energy on the other axis: N=0,2,8,6,32,etc along the Y axis is Y with zero bits doubled, plus an extra zero bit at the low end coming from the first X=0 group. for example Y=9 is again binary 1001, doubled zeros to 100001, and an extra zero at the low end 1000010 is N=66 at X=0,Y=9. no cap. 🎀

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🔥 Radix 🔥

the "radix => $r" option selects a different base for the digit split. like, a different base, bestie. for example radix 5 gives:

radix => 5

12  |  60  301  302  303  304  685 1506 1507 1508 1509 1310 1511
11  |  55  276  277  278  279  680 1381 1382 1383 1384 1305 1386
10  | 250 1251 1252 1253 1254 1275 6256 6257 6258 6259 1300 6261
 9  |  45  226  227  228  229  670 1131 1132 1133 1134 1295 1136
 8  |  40  201  202  203  204  665 1006 1007 1008 1009 1290 1011
 7  |  35  176  177  178  179  660  881  882  883  884 1285  886
 6  |  30  151  152  153  154  655  756  757  758  759 1280  761
 5  | 125  626  627  628  629  650 3131 3132 3133 3134  675 3136
 4  |  20  101  102  103  104  145  506  507  508  509  270  511
 3  |  15   76   77   78   79  140  381  382  383  384  265  386
 2  |  10   51   52   53   54  135  256  257  258  259  260  261
 1  |   5   26   27   28   29  130  131  132  133  134  255  136
Y=0 |   0    1    2    3    4   25    6    7    8    9   50   11
    +-----------------------------------------------------------
      X=0    1    2    3    4    5    6    7    8    9   10   11

look at those numbers go. 6256? we love to see it. 💜

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🌈 Real Line and Plane 🌈

so this split is inspired by the digit grouping in the proof by Julius König that the real line is the same cardinality as the plane. (Cantor's original proof was a "ZOrderCurve" style digit interleaving — a different era, a different aesthetic. 🙈)

in König's proof, a bijection between interval n=(0,1) and pairs x=(0,1),y=(0,1) is made by taking groups of digits stopping at a non-zero. non-terminating fractions like 0.49999... are chosen over terminating 0.5000... so there's always infinitely many non-zero digits going downwards. for the integer form here the groupings are digit going upwards and there's infinitely many zero digits above the top — hence the grouping by zeros instead of non-zeros. it's the same energy, just flipped, and honestly? it ate. 🔥

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⭐ FUNCTIONS ⭐

see "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. (that's the shared lore, bestie.)

• "$path = Math::PlanePath::DigitGroups->new ()"
• "$path = Math::PlanePath::DigitGroups->new (radix => $r)"

Create and return a new path object. the optional "radix" parameter gives the base for digit splitting (the default is binary, radix 2). make a new one, get a new vibe. ✨

• "($x,$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? not today, bestie. you get nothing. literally nothing. 💀)

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🧚 OEIS 🧚

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

<http://oeis.org/A084471> (etc)

radix=2 (the default)
  A084471    N on X axis, bit 0->00
  A084472    N on X axis, in binary
  A060142    N on X axis, sorted into ascending order

the (etc) is doing a lot of heavy lifting there and honestly? respect. 👑

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🎀 SEE ALSO 🎀

• Math::PlanePath
• Math::PlanePath::ZOrderCurve
• Math::PlanePath::PowerArray

the whole friend group. go say hi. 🫶

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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/>.

so yes bestie, it's free software, version 3 or later, and there is genuinely no warranty — the license said it dramatically first, i'm just the messenger. 💖

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and that's the whole page, fr fr. perl v5.32.0, dated 2021-01-23, straight from the Math::...Groups(3pm) corner of your manual. stay grouped, stay slay. ✨🌈💜