Math::...:MPeaks(3pm)

💖 Math::...:MPeaks(3pm) 💖

okay bestie, welcome to the User Contributed Perl Documentation for Math::...:MPeaks(3pm) — yes the manual name looks like someone's cat walked across the keyboard, but the module itself is Math::PlanePath::MPeaks and it's adorable. sit down, we're doing geometry 💅

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✨ NAME ✨

Math::PlanePath::MPeaks — points in expanding M shape 🦋

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🌸 SYNOPSIS 🌸

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

that's it. three lines. you're already iconic.

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🎀 DESCRIPTION 🎀

so this path literally puts points in layers of an "M" shape, no cap:

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

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

k so N=1 to N=5 is the first "M" shape, then N=6 to N=16 sits on top of that, etc. The centre goes half way down. And if we reckon N=1 to N=5 as layer d=1, then bestie, hold onto your calculator:

Xleft = -d
Xright = d
Ypeak = 2*d - 1
Ycentre = d - 1

Each "M" is 6 points longer than the preceding one. The verticals are each 2 longer, and the centre diagonals each 1 longer. That step-6 situation is giving major "HexSpiral" energy, iykyk.

The octagonal numbers N=1,8,21,40,65,etc k*(3k-2) are a straight line of slope 2 going up to the left. And the octagonal numbers of the second kind N=5,16,33,56,etc k*(3k+2) are along the X axis to the right. ate. 🔥

🌈 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

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

look at that, 0 in the middle, main character energy ✨

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🔥 FUNCTIONS 🔥

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. (translation: all the path classes share these, go read the family docs too 🫶)

• $path = Math::PlanePath::MPeaks->new () — Create and return a new path object. brand new baby path, obsessed.
• ($x,$y) = $path->n_to_xy ($n) — Return the X,Y coordinates of point number $n on the path.

For "$n < 0.5" the return is an empty list, it being considered there are no negative points. so like, nothing there, sorry bestie 🙈

• $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 points as a squares of side 1, so the half-plane y>=-0.5 is entirely covered.

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👑 OEIS 👑

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

<http://oeis.org/A045944> (etc)

n_start=1 (the default)
  A045944    N on X axis >= 1, extra initial 0
               being octagonal numbers second kind
  A056106    N on Y axis, extra initial 1
  A056109    N on X negative axis <= -1

n_start=0
  A049450    N on Y axis, extra initial 0, 2*pentagonal

n_start=2
  A027599    N on Y axis, extra initial 6,2

go off, sequence database 💀

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💜 SEE ALSO 💜

Math::PlanePath, Math::PlanePath::PyramidSides

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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/>.

okay the all-caps bits are the legal ones so we keep them loud, but fr fr it's GPL v3-or-later, go read it, it's giving freedom 💖

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and that's the whole page, bestie 🫶 this one shipped with perl v5.32.0 and was dated 2021-01-23. slay, go plot some M shapes. periodt. 💅✨