Math::...veStair(3pm)
🎀 NAME 🎀
Hiii bestie 💖 welcome to the Math::...veStair(3pm) page — yep, that's the User Contributed Perl Documentation corner of the internet, and the star of the show is Math::PlanePath::SierpinskiCurveStair, aka the Sierpinski curve but make the diagonals stair-step. That's it, that's the whole assignment. 👑
✨ SYNOPSIS ✨
use Math::PlanePath::SierpinskiCurveStair;
my $path = Math::PlanePath::SierpinskiCurveStair->new (arms => 2);
my ($x, $y) = $path->n_to_xy (123);💖 DESCRIPTION 💖
Okay so this is a remix of the "SierpinskiCurve", except every diagonal bit gets chopped into little stair steps — horizontal then vertical, no shortcuts. Look at her go 😭💅
10 | 52-53
| | |
9 | 50-51 54-55
| | |
8 | 49-48 57-56
| | |
7 | 42-43 46-47 58-59 62-63
| | | | | | |
6 | 40-41 44-45 60-61 64-65
| | |
5 | 39-38 35-34 71-70 67-66
| | | | | | |
4 | 12-13 37-36 33-32 73-72 69-68 92-93
| | | | | | |
3 | 10-11 14-15 30-31 74-75 90-91 94-95
| | | | | | |
2 | 9--8 17-16 29-28 77-76 89-88 97-96
| | | | | | |
1 | 2--3 6--7 18-19 22-23 26-27 78-79 82-83 86-87 98-99
| | | | | | | | | | | | |
Y=0 | 0--1 4--5 20-21 24-25 80-81 84-85 ...
|
+-------------------------------------------------------------
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19The tiling? Literally the same as the "SierpinskiCurve", bestie — only the diagonals become a stair step, one horizontal plus one vertical. The correspondence goes like:
SierpinskiCurve SierpinskiCurveStair
7-- 12--
/ |
6 10-11
| |
5 9--8
\ |
1--2 4 2--3 6--7
/ \ / | | |
0 3 0--1 4--5So the "SierpinskiCurve" N=0 to N=3 maps onto N=0 to N=5 over here. Then N=7 to N=12 is a whole copy of that N=0 to N=5 base. And point N=6? That's the little extra sitting between the parts. The next extra one of those is N=19. 🙈
🦋 Diagonal Length 🦋
The "diagonal_length" option lets you make the diagonals longer — still stair-step styled, obviously, we don't break the theme. Like:
diagonal_length => 4
10 | 36-37
| | |
9 | 34-35 38-39
| | |
8 | 32-33 40-41
| | |
7 | 30-31 42-43
| | |
6 | 28-29 44-45
| | |
5 | 27-26 47-46
| | |
4 | 8--9 25-24 49-48 ...
| | | | | |
3 | 6--7 10-11 23-22 51-50 62-63
| | | | | |
2 | 4--5 12-13 21-20 53-52 60-61
| | | | | |
1 | 2--3 14-15 18-19 54-55 58-59
| | | | | |
Y=0 | 0--1 16-17 56-57
|
+------------------------------------------------------
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17The length is reckoned from N=0 to the end of the first side at N=8, which runs X=1 to X=5 when the length is 4 units. Math is mathing. ✨
🌈 Arms 🌈
The optional "arms" parameter can give you up to eight copies of the curve, each one advancing successively. Eight! The audacity. 😭 For example:
arms => 8
98-90 66-58 57-65 89-97 5
| | | | | |
99 82-74 50-42 41-49 73-81 96 4
| | | |
91-83 26-34 33-25 80-88 3
| | | |
67-75 18-10 9-17 72-64 2
| | | |
59-51 27-19 2 1 16-24 48-56 1
| | | | | |
43-35 11--3 . 0--8 32-40 <- Y=0
44-36 12--4 7-15 39-47 -1
| | | | | |
60-52 28-20 5 6 23-31 55-63 -2
| | | |
68-76 21-13 14-22 79-71 -3
| | | |
92-84 29-37 38-30 87-95 -4
| |
85-77 53-45 46-54 78-86 -5
| | | | | |
93 69-61 62-70 94 -6
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6The multiples of 8 (or however many arms you picked) — N=0, 8, 16, etc — that's the original curve, and everything mod 8 beyond that is one of the copies.
The little . in the middle there is the origin at X=0,Y=0. It would be more symmetrical to shove the origin into the middle of the eight arms (that'd be X=-0.5,Y=-0.5 in the picture above), but nooo, that would hand you fractional X,Y values and we don't do that here. Apply an offset of X+0.5,Y+0.5 if you want it centred. Slay. 💅
👑 Level Ranges 👑
N=0 is reckoned as level=0, then N=0 to N=5 inclusive is level=1, and so on up forever. Each level is 4 copies of the previous one, plus 2 extra points in between. 💀
LevelPoints[k] = 4*LevelPoints[k-1] + 2 starting LevelPoints[0]=1
= 2 + 2*4 + 2*4^2 + ... + 2*4^(k-1) + 1*4^k
= (5*4^k - 2)/3
Nlevel[k] = LevelPoints[k] - 1 since starting at N=0
= 5*(4^k - 1)/3
= 0, 5, 25, 105, 425, 1705, 6825, 27305, ... (A146882)The width along the X axis of a level doubles each time, plus an extra distance of 3 in between:
LevelWidth[k] = 2*LevelWidth[k-1] + 3 starting LevelWidth[0]=0
= 3 + 3*2 + 3*2^2 + ... + 3*2^(k-1) + 0*2^k
= 3*(2^k - 1)
Xlevel[k] = 1 + LevelWidth[k]
= 3*2^k - 2
= 1, 4, 10, 22, 46, 94, 190, 382, ... (A033484)🧚 Level Ranges with Diagonal Length 🧚
When "diagonal_length" = L, level=0 is reckoned as having L many points instead of just the one:
LevelPoints[k] = 2 + 2*4 + 2*4^2 + ... + 2*4^(k-1) + L*4^k
= ( (3L+2)*4^k - 2 )/3
Nlevel[k] = LevelPoints[k] - 1
= ( (3L+2)*4^k - 5 )/3And the width of level=0 becomes L-1 instead of 0:
LevelWidth[k] = 2*LevelWidth[k-1] + 3 starting LevelWidth[0]=L-1
= 3 + 3*2 + 3*2^2 + ... + 3*2^(k-1) + (L-1)*2^k
= (L+2)*2^k - 3
Xlevel[k] = 1 + LevelWidth[k]
= (L+2)*2^k - 2Think of level=0 with L many points as a cute little block that gets replicated in mirror image to build level=1. Like, the diagonal 4 example from earlier turns into:
8 9 diagonal_length => 4
| |
6--7 10-11
| |
. 5 12 .
2--3 14-15
| |
0--1 16-17The spacing between the parts in the tiling comes from taking a margin of 1/2 at the base and 1 horizontally on the left and right. 🫶
🍓 Level Fill 🍓
Plot twist: the curve does not visit all the points in that eighth of the plane below the X=Y diagonal. Not even close. In general, Nlevel+1 many points get visited out of the triangular area Xlevel^2/4, so the filled fraction approaches a constant:
Nlevel 4*(3L+2)
FillFrac = ------------ -> ---------
Xlevel^2 / 4 3*(L+2)^2For example, the default L=1 gives FillFrac=20/27=0.74. Or L=2 gives FillFrac=2/3=0.66. As the diagonal length grows, that fraction shrinks — because the holes in the pattern keep getting bigger. It's giving "more curve, more gaps". 🌸
🛠 FUNCTIONS 🛠
Go peek at "FUNCTIONS" in Math::PlanePath for all the behaviour that's common to every path class. 💜
$path = Math::PlanePath::SierpinskiCurveStair->new ()$path = Math::PlanePath::SierpinskiCurveStair->new (diagonal_length => $L, arms => $A)— create and return a shiny new path object, bestie.($x,$y) = $path->n_to_xy ($n)— returns the X,Y coordinates of point number$non the path. Points start at 0, and if"$n < 0"then you get an empty list back, sorry. 😭
Fractional positions give you an X,Y position along a straight line between the integer positions.
$n = $path->n_start()— returns 0, the first N in the path. Periodt.
🎀 Level Methods 🎀
($n_lo, $n_hi) = $path->level_to_n_range($level)— returns(0, ((3*$diagonal_length +2) * 4**$level - 5)/3exactly as per"Level Ranges with Diagonal Length"up above. No cap. 💅
🔢 OEIS 🔢
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include:
<http://oeis.org/A146882> (etc)
A146882 Nlevel, for level=1 up
A033484 Xmax and Ymax in level, being 3*2^n - 2👀 SEE ALSO 👀
Math::PlanePathMath::PlanePath::SierpinskiCurve
🏠 HOME PAGE 🏠
<http://user42.tuxfamily.org/math-planepath/index.html>⚖️ LICENSE ⚖️
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
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/>.
ngl bestie, that warranty disclaimer is doing the most, but read it — no promises, no guarantees, that's the deal. 🙈
okay that's the whole page, we ate and left zero crumbs 👑✨ this was perl v5.32.0, dated 2021-01-23, and it's the Math::...veStair(3pm) page — love that for us, bye bestie 💖🦋🌸
📦 where this page comes from
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