Math::...Octant(3pm)
hey bestie 💖 welcome to Math::...Octant(3pm) from User Contributed Perl Documentation 🎀 okay so this is the math-plane-path girlies' corner of the internet and this particular one is lowkey gorgeous. geometry? slay. let's get into it 🦋
✨ NAME ✨
Math::PlanePath::HypotOctant — it's literally the octant of points, ordered by hypotenuse distance 💫
🌸 SYNOPSIS 🌸
use Math::PlanePath::HypotOctant;
my $path = Math::PlanePath::HypotOctant->new;
my ($x, $y) = $path->n_to_xy (123);🎀 DESCRIPTION 🎀
okay so this path walks an octant of integer points X,Y and it visits them in order of how far they are from the origin 0,0 🌈 the points make a rising triangle where 0<=Y<=X, and it looks like this:
8 | 61
7 | 47 54
6 | 36 43 49
5 | 27 31 38 44
4 | 18 23 28 34 39
3 | 12 15 19 24 30 37
2 | 6 9 13 17 22 29 35
1 | 3 5 8 11 16 21 26 33
Y=0 | 1 2 4 7 10 14 20 25 32 ...
+---------------------------------------
X=0 1 2 3 4 5 6 7 8like, for example, N=11 at X=4,Y=1 is sqrt(4*4+1*1) = sqrt(17) from the origin 🧮 and the next furthest away from the origin is X=3,Y=3 at sqrt(18). obsessed with the tidiness fr fr.
this octant is "primitive" elements X^2+Y^2 in the sense that it excludes negative X or Y or swapped Y,X 💅 (so it's the no-duplicates, no-negatives version, iykyk)
🔥 Equal Distances 🔥
when points have the same distance from the origin, they get taken in anti-clockwise order starting from the X axis — which, translation, means by increasing Y. and points only end up with the same distance when there's more than one way to write that distance as the sum of two squares 👑
Pythagorean triples hand you a point on the X axis and a point above it. for example 5^2 == 4^2 + 3^2 has N=14 at X=5,Y=0 which is just plain old 5^2 = 5^2 + 0, and then N=15 at X=4,Y=3 for the triple. both of them are 5 away from the origin. iconic behaviour honestly 💖
and combinations like 20^2 + 15^2 == 24^2 + 7^2 happen too 😭 and it can even go off with three or more different ways to land on the same sum distance. the math is mathing.
🦋 Even Points 🦋
the option points => "even" visits just the even points, meaning the sum X+Y is even — so X,Y are both even or both odd ✨
12 | 66
11 | points => "even" 57
10 | 49 58
9 | 40 50
8 | 32 41 51
7 | 25 34 43
6 | 20 27 35 45
5 | 15 21 29 37
4 | 10 16 22 30 39
3 | 7 11 17 24 33
2 | 4 8 13 19 28 38
1 | 2 5 9 14 23 31
Y=0 | 1 3 6 12 18 26 36
+---------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12the even points can be mapped onto all the points with a 45 degree rotate and flip, which is so cute. N=1,3,6,12,etc on the X axis here is on the X=Y diagonal of all-points. and conversely, N=1,2,4,7,10,etc on the X=Y diagonal here is on the X axis of all-points 💫
all_X = (even_X + even_Y) / 2
all_Y = (even_X - even_Y) / 2
even_X = (all_X + all_Y)
even_Y = (all_X - all_Y)the sets of points with equal hypotenuse are the same in the even version and the all version, but the flip takes them in reverse order 👀 the first such reversal shows up at N=14 and N=15. in even-points they're at 7,1 and 5,5. in all-points they're at 5,0 and 4,3, and those two map to 5,5 and 7,1 — ie. the opposite way around. not me getting emotional over a coordinate flip 😭
💜 Odd Points 💜
the option points => "odd" visits just the odd points, meaning the sum X+Y is odd — so X,Y are one odd and the other even 🌙
12 | 66
11 | points => "odd" 57
10 | 47 58
9 | 39 49
8 | 32 41 51
7 | 25 33 42
6 | 20 26 35 45
5 | 14 21 29 37
4 | 10 16 22 30 40
3 | 7 11 17 24 34
2 | 4 8 13 19 28 38
1 | 2 5 9 15 23 31
Y=0 | 1 3 6 12 18 27 36
+------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13the X=Y diagonal gets excluded here because it has X+Y even 🙈 sorry diagonal, not your era.
🌟 FUNCTIONS 🌟
peep "FUNCTIONS" in Math::PlanePath for the behaviour that's common to literally every path class, no cap.
$path = Math::PlanePath::HypotOctant->new ()$path = Math::PlanePath::HypotOctant->new (points => $str)— make a new hypot octant path object and hand it back to you 🫶 thepointsoption can be:
"all" all integer X,Y (the default)
"even" only points with X+Y even
"odd" only points with X+Y odd($x,$y) = $path->n_to_xy ($n)— returns the X,Y coordinates of point number$non the path ✨- for
$n < 1you get back an empty list, because the first point at X=0,Y=0 is considered to be N=1 💫 - right now it's unspecified what happens if
$nisn't an integer. the successive points are a fair way apart, so handing it an X,Y position in between the integer$nmight not make much sense 🤷♀️
- for
$n = $path->xy_to_n ($x,$y)— returns an integer point number for the coordinates$x,$y. every integer N is treated as the centre of a unit square, and an$x,$ythat lands inside that square returns that N 🎯
🧮 FORMULAS 🧮
ngl the calculations are not very efficient currently 💀 for each Y row, a current X and the matching hypotenuse X^2+Y^2 are kept around. to find the next furthest, it searches through all those hypotenuses looking for the smallest — including the equal smallest ones — and those become the next N points ✨
for n_to_xy() an array gets built inside the object, and it's used for repeat calculations. for xy_to_n() an array of hypot to N gives you a the first N of a given X^2+Y^2 distance, and then a search runs through the next few N for the case where there's more than one X,Y at that hypot 👑
🍓 OEIS 🍓
entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include:
<http://oeis.org/A024507> (etc)
points="all"
A024507 X^2+Y^2 of all points not on X axis or X=Y diagonal
A024509 X^2+Y^2 of all points not on X axis
being integers occurring as sum of two non-zero squares,
with repetitions for multiple ways
points="even"
A036702 N on X=Y leading Diagonal
being count of points norm<=k
points="odd"
A057653 X^2+Y^2 values occurring
ie. odd numbers which are sum of two squares,
without repetitions👑 SEE ALSO 👑
Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::TriangularHypot, Math::PlanePath::PixelRings, Math::PlanePath::PythagoreanTree — the whole squad, go say hi 🫶
🌈 HOME PAGE 🌈
<http://user42.tuxfamily.org/math-planepath/index.html>
💖 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 do not skip the license part, it's the legal main character energy and it stays, periodt.
and that's a wrap! this lil number came from perl v5.32.0 dated 2021-01-23 — shoutout to the Math::...Octant(3pm) page, you ate and left no crumbs 💅✨
📦 where this page comes from
- debian bookworm: libmath-planepath-perl 129-1 · GPL-3.0-or-later
- debian trixie: libmath-planepath-perl 129-1 · GPL-3.0-or-later
- ubuntu jammy: libmath-planepath-perl 129-1 · GPL-3.0-or-later
- ubuntu noble: libmath-planepath-perl 129-1 · GPL-3.0-or-later
- ubuntu resolute: libmath-planepath-perl 129-1 · GPL-3.0-or-later
- ubuntu stonking: libmath-planepath-perl 129-1 · GPL-3.0-or-later
- 💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on : a language model's rewrite of the original, under its terms