Math::...idSides(3pm)

💖 Math::...idSides — aka Math::PlanePath::PyramidSides 💖

okay bestie so this is a page from the User Contributed Perl Documentation manual, and it's about a lil module called Math::PlanePath::PyramidSides — a plane path that walks points up along the sides of a pyramid. it's giving geometry class but make it cute. let's get into it 🫶

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✨ NAME ✨

Math::PlanePath::PyramidSides — points along the sides of pyramid. yes that's the whole name, she's lowkey understated like that 💅

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🌸 SYNOPSIS 🌸

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

literally that's it. three lines and you're living your best life 🦋

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👑 DESCRIPTION 👑

okay so this path lays points down in layers along the sides of a pyramid, and the pyramid grows upward. main character energy tbh:

                               21                        4
                           20  13  22                    3
                       19  12   7  14  23                2
                   18  11   6   3   8  15  24            1
               17  10   5   2   1   4   9  16  25    <- Y=0
              ------------------------------------
                                ^
           ... -4  -3  -2  -1  X=0  1   2   3   4 ...

so like, N=1,4,9,16,etc along the positive X axis? those are the perfect squares, no cap. and N=2,6,12,20,etc sitting in the X=-1 vertical are the pronic numbers k*(k+1) — the halfway point between two successive squares. cute lil midpoint moment 🎀

the pattern is the same as the "Corner" path, but turned and spread out so that the single quadrant in "Corner" becomes a half-plane here. glow up fr.

and it's similar to "PyramidRows" (with its default step=2), except the columns get dropped down vertically to start at the X axis. any pattern inside a column stays exactly the same, but what used to be a row becomes a diagonal, and vice versa. slay 🔥

🍓 Lucky Numbers of Euler 🍓

an interesting sequence for this path is Euler's k^2+k+41. the low values are spread around a bit, but from N=1763 (k=41) they're the vertical at X=40. there are quite a few primes in this quadratic, and when you plot primes, that vertical stands out a little denser than its surroundings (at least for up to the first 2500 or so values, iykyk). the line shows up in other step==2 paths too, but not as clearly — in the "PyramidRows" for instance the beginning is up at Y=40, and in the "Corner" path it's a diagonal.

🧚 N Start 🧚

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

n_start => 0

and then it looks like:

                       20                    4
                    19 12 21                 3
                 18 11  6 13 22              2
              17 10  5  2  7 14 23           1
           16  9  4  1  0  3  8 15 24    <- Y=0
           --------------------------
           -4 -3 -2 -1 X=0 1  2  3  4

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💅 FUNCTIONS 💅

check "FUNCTIONS" in Math::PlanePath for the behaviour that's common to all the path classes, bestie.

• $path = Math::PlanePath::PyramidSides->new ()
• $path = Math::PlanePath::PyramidSides->new (n_start => $n) — create and return a new path object. brand new path, who dis 💖
• ($x,$y) = $path->n_to_xy ($n) — return the X,Y coordinates of point number $n on the path. and heads up: for "$n < 0.5" the return is an empty list, because the vibe is that there are no negative points in the pyramid. sorry babe 🙈
• $n = $path->xy_to_n ($x,$y) — return the point number for coordinates "$x,$y". $x and $y each get rounded to the nearest integer, which has the effect of treating points in the pyramid as squares of side 1, so the half-plane y>=-0.5 is entirely covered. full coverage, periodt ✨
• ($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2) — the returned range is exact, meaning $n_lo and $n_hi are literally the smallest and biggest in the rectangle. no guesswork, no crumbs.

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🌈 FORMULAS 🌈

🌟 Rectangle to N Range 🌟

for "rect_to_n_range()", in each column N increases, so the biggest N lives in the topmost row and the smallest N lives in the bottom row.

in each row N increases along the sequence X=0,-1,1,-2,2,-3,3, etc. so the biggest N is at the X with the biggest absolute value, and it prefers the positive X=k over the negative X=-k. biased much? 😭

the smallest N, conversely, is at the X with the smallest absolute value. and if the X range crosses 0 — ie. $x1 and $x2 have different signs — then X=0 is the smallest. crossing zero is kind of a moment ✨

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🔥 OEIS 🔥

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

<http://oeis.org/A196199> (etc)

n_start=1 (the default)
  A049240    abs(dY), being 0=horizontal step at N=square
  A002522    N on X negative axis, x^2+1
  A033951    N on X=Y diagonal, 4d^2+3d+1
  A004201    N for which X>=0, ie. right hand half
  A020703    permutation N at -X,Y

n_start=0
  A196199    X coordinate, runs -n to +n
  A053615    abs(X), runs n to 0 to n
  A000196    abs(X)+abs(Y), being floor(sqrt(N)),
               k repeated 2k+1 times starting 0

number nerds stay winning 🧮💜

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🦋 SEE ALSO 🦋

go check out her friends, they're all iconic:

• Math::PlanePath
• Math::PlanePath::PyramidRows
• Math::PlanePath::Corner
• Math::PlanePath::DiamondSpiral
• Math::PlanePath::SacksSpiral
• Math::PlanePath::MPeaks

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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/>.

okay real talk for one sec: that "without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE" bit is ALL CAPS FOR A REASON. it means if it breaks, that's on you, not on Kevin. dramatic but accurate 💀

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and that's the whole page, bestie! 🎀 this lil beauty shipped as part of the User Contributed Perl Documentation, perl v5.32.0, dated 2021-01-23. go forth and plot pyramids ✨👑💖