Math::...perSide(3pm)

📦

version 129 · 3 versions

129 · debian bookworm, debian trixie, ubuntu jammy, ubuntu noble, ubuntu resolute, ubuntu stonking (latest)
129 · fedora 43
129 · fedora 44

, in debian bookworm, debian trixie, ubuntu jammy, ubuntu noble, ubuntu resolute, ubuntu stonking

💖 Math::PlanePath::GosperSide 💖

okay bestie, gather round, because this is the manpage for Math::PlanePath::GosperSide — literally one single side of the Gosper island, and it's giving geometry main character energy. (The man page itself shows up as Math::...perSide(3pm) when it's feeling shy, but we know her.) 🦋

✨ NAME ✨

Math::PlanePath::GosperSide — one side of the Gosper island. that's it, that's the whole vibe. 🏝️

🌸 SYNOPSIS 🌸

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

look at her. three lines. no cap. 🎀

🦋 DESCRIPTION 🦋

So this path is a single side of the Gosper island, but like — in integers. ("Triangular Lattice" in Math::PlanePath, iykyk.)

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

            ^
           X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 ...

the path slowly spirals around counter clockwise, with a LOT of wiggling in between. it's not a clean spiral, bestie, it's a wiggle queen. 💅 The N=3^level point lands at

N = 3^level
angle = level * atan(sqrt(3)/5)
      = level * 19.106 degrees
radius = sqrt(7) ^ level

a full revolution for example takes roughly level=19, which is about N=1,162,000,000. yes, billion. the numbers are number-ing. 😭

Both ends of such levels are in fact sub-spirals, like an "S" shape. slay.

The path is both the sides and the radial spokes of the "GosperIslands" path, as described in "Side and Radial Lines" in Math::PlanePath::GosperIslands. Each N=3^level point is the start of a "GosperIslands" ring. obsessed with the interconnection honestly. ✨

The path is the same as the "TerdragonCurve" except the turns here are by 60 degrees each, whereas "TerdragonCurve" is by 120 degrees. See Math::PlanePath::TerdragonCurve for the turn sequence and total direction formulas etc. sibling energy fr fr. 🫶

💜 FUNCTIONS 💜

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. (yes there's a whole shared foundation, we're not special, we're just her.)

$path = Math::PlanePath::GosperSide->new () — create and return a new path object. brand new, fresh out the box, iconic. 👑
($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. (negativity not welcome here 🙈)

Fractional $n gives a point on the straight line between integer N. fancy interpolation moment.

🌈 Level Methods 🌈

($n_lo, $n_hi) = $path->level_to_n_range($level) — return (0, 3**$level). that's the whole range, no notes. ⭐

🧚 FORMULAS 🧚

🎀 Level Endpoint 🎀

The endpoint of each level N=3^k is at

X + Y*i*sqrt(3) = b^k
where b = 2 + w = 5/2 + sqrt(3)/2*i
      where w=1/2 + sqrt(3)/2*i sixth root of unity

X(k) = ( 5*X(k-1) - 3*Y(k-1) )/2        for k>=1
Y(k) = (   X(k-1) + 5*Y(k-1) )/2
       starting X(0)=2 Y(0)=0

X(k) = 5*X(k-1) - 7*X(k-2)        for k>=2
       starting X(0)=2 X(1)=5
     = 2, 5, 11, 20, 23, -25, -286, -1255, -4273, -12580, -32989,..

Y(k) = 5*Y(k-1) - Y*X(k-2)        for k>=2
       starting Y(0)=0 Y(1)=1
     = 0, 1,  5, 18, 55, 149,  360,   757,  1265, 1026, -3725, ...
                                                        (A099450)

like the sixth root of unity? the math is mathing. 🍓

The curve base figure is XY(k)=XY(k-1)+rot60(XY(k-1))+XY(k-1) giving XY(k) = (2+w)^k = b^k where w is the sixth root of unity giving the rotation by +60 degrees.

The mutual recurrences are similar with the rotation done by (X-3Y)/2, (Y+X)/2 per "Triangular Lattice" in Math::PlanePath. The separate recurrences are found by using the first to get Y(k-1) = -2/3*X(k) + 5/3*X(k-1) and substitute into the other to get X(k+1). Similar the other way around for Y(k+1). it's algebra o'clock and we're thriving. 💀

⭐ OEIS ⭐

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

<http://oeis.org/A099450> (etc)

A229215   direction 1,2,3,-1,-2,-3 (clockwise)
A099450   Y at N=3^k (for k>=1)

Also the turn sequence is the same as the terdragon curve, see "OEIS" in Math::PlanePath::TerdragonCurve for the several turn forms, N positions of turns, etc. cross-referencing like a boss. 👑

🔥 SEE ALSO 🔥

Math::PlanePath, Math::PlanePath::GosperIslands, Math::PlanePath::TerdragonCurve, Math::PlanePath::KochCurve

Math::Fractal::Curve

(the whole friend group, all iconic, all linked 💖)

🌸 HOME PAGE 🌸

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

go say hi. 🦋

💅 LICENSE 💅

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 read the license, it's legally binding and also it's giving "free software, free love, no warranty whatsoever" 💀

and that's the whole page!! 🎀 this was generated from perl v5.32.0, dated 2021-01-23, and the man page answers to Math::...perSide(3pm). mwah, go plot some spirals, you absolute legend. 💖✨🦋👑

📦 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 2026-10-14: a language model's rewrite of the original, under its terms