Math::...lSkewed(3pm)

💖 NAME 💖

okay bestie, pull up a chair because today we're talking about Math::PlanePath::PentSpiralSkewed — that's the module that draws integer points in a pentagonal shape 🦋✨ (yes it's from the User Contributed Perl Documentation, and yes it's serving main character energy).

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🌸 SYNOPSIS 🌸

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

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🎀 DESCRIPTION 🎀

so this path makes a pentagonal (five-sided) spiral, except the points are skewed so they can squeeze into a square grid and fully cover the plane. it's giving "geometry but make it fit" 💅

      10 ...             2
     /  \  \
   11  3  9 20           1
  /  /  \  \  \
12  4  1--2  8 19    <- Y=0
  \  \       |  |
   13  5--6--7 18       -1
     \          |
      14-15-16-17       -2

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

the pattern is lowkey similar to the "SquareSpiral", but it cuts three corners which makes each cycle is faster. each cycle is literally just 5 steps longer than the previous one (where it's 8 for a "SquareSpiral"). no cap, that's the whole glow-up 🔥

✨ N Start ✨

the default is to number points starting at N=1 like you see up above 👆 an optional "n_start" can give a different start, in the same pattern. for example, to start at 0:

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

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🫶 FUNCTIONS 🫶

peep "FUNCTIONS" in Math::PlanePath for the behaviour that's common to all path classes — this one's just doing its own little thing on top 🧚

• "$path = Math::PlanePath::PentSpiral->new ()" — creates and returns a brand new path object. fresh out the box, iconic ✨
• "$path = Math::PlanePath::PentSpiral->new (n_start => $n)" — same thing, except you get to pick where the numbering starts 🎀
• "$n = $path->xy_to_n ($x,$y)" — returns the point number for coordinates "$x,$y". both $x and $y get rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1 📐

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🌈 OEIS 🌈

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

<http://oeis.org/A192136> (etc)

n_start=1 (the default)
  A192136    N on X axis, (5*n^2 - 3*n + 2)/2
  A140066    N on Y axis
  A116668    N on X negative axis, (5n^2 + n + 2)/2
  A134238    N on Y negative axis
  A158187    N on North-West diagonal, 10*n^2 + 1
  A005891    N on South-East diagonal, centred pentagonals

n_start=0
  A000566    N on X axis, heptagonal numbers
  A005476    N on Y axis
  A005475    N on X negative axis
  A147875    N on Y negative axis, second heptagonals
  A033583    N on North-West diagonal, 10*n^2
  A028895    N on South-East diagonal, 5*triangular

look at all those sequence numbers bestie, it's a whole buffet 🍓

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💜 SEE ALSO 💜

• Math::PlanePath
• Math::PlanePath::SquareSpiral
• Math::PlanePath::DiamondSpiral
• Math::PlanePath::HexSpiralSkewed

go read the whole squad, they're all slaying in their own lanes 👑

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🏠 HOME PAGE 🏠

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

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👑 LICENSE 👑

Copyright 2010, 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/>.

ok so like, read the license fr fr. it's not a vibe, it's a legal thing 🙈

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and that's the whole page, bestie!! this one shipped as perl v5.32.0, dated 2021-01-23, and it lives under the name Math::...lSkewed(3pm) in your friendly local perl install 💅✨ mwah, go spiral responsibly 🌀💖