Math::...Tiling(3pm)
🌸 Math::...Tiling 🌸
okay bestie, so the full name of this page is Math::...Tiling(3pm) — it's the User Contributed Perl Documentation and it lives in Math::PlanePath::LTiling 💖
✨ NAME ✨
Math::PlanePath::LTiling — a 2x2 self-similar tiling made of four pattern parts 💅
🦋 SYNOPSIS 🦋
use Math::PlanePath::LTiling;
my $path = Math::PlanePath::LTiling->new;
my ($x, $y) = $path->n_to_xy (123);👑 DESCRIPTION 👑
So this is a self-similar tiling made out of "L" shapes, and it's genuinely so cute. You take a base "L", then replicate it four times with the end parts turned +90 and -90 degrees to make a bigger L. Like this:
+-----+-----+
|12 | 15|
| +--+--+ |
| |14 | |
+--+ +--+--+
| | |11 |
| +--+ +--+
|13 | | |
+-----+ +-----+--+ +--+--+-----+
| 3 | | 3 | |10 | | 5|
| +--+ --> | +--+ +--+--+ +--+ |
| | | | | | 8 | 9 | | |
+--+ +--+ +--+--+ +--+ +--+--+--+--+ +--+
| | --> | | 2 | | | | 2 | | | 6 | |
| +--+ | +--+--+ | | +--+--+ | +--+--+ |
| 0 | | 0 | 1 | | 0 | 1 | 7 | 4 |
+-----+ +-----+-----+ +-----+-----+-----+-----+The parts get numbered left, then middle, then upper. And that relative numbering stays put when it gets rotated at the next replication level — think N=4 to N=7, no cap.
What you end up with is 1 of every 3 points in the first quadrant, with this subtle little layout of points and spaces that makes diagonal lines and teeny 2x2 blocks. Obsessed honestly:
15 | 48 51 61 60 140 143 163
14 | 50 62 142 168
13 | 56 59 139 162
12 | 49 58 63 141 160
11 | 55 44 47 131 138
10 | 57 46 136 137
9 | 54 43 130 134
8 | 52 53 45 128 129 135
7 | 12 15 35 42 37 21
6 | 14 40 41 22
5 | 11 34 38 25
4 | 13 32 33 39 36
3 | 3 10 5 31 26
2 | 8 9 27 24
1 | 2 6 30 18
Y=0 | 0 1 7 4 28 29 19
+------------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14On the X=Y leading diagonal, N=0,2,8,10,32,etc — those are the integers made from only digits 0 and 2 in base 4. Or, same energy, the integers which have zero bits at all even numbered positions, binary c0d0e0f0 ⭐
🌈 Left or Upper 🌈
Option L_fill => "left" or L_fill => "upper" numbers the tiles at their left end or upper end respectively. Slay, let's look:
L_fill => 'left' 8 | 52 45 43
7 | 15 42
+-----+ 6 | 12 35 40
| | 5 | 14 34 33
| +--+ 4 | 13 11 32
| 3| | 3 | 10 9 5
+--+ +--+--+ 2 | 3 8 6 31
| | 2| 1| 1 | 2 1 4
| +--+--+ | Y=0 | 0 7
| 0| | +------------------------------------
+-----+-----+ X=0 1 2 3 4 5 6 7 8
L_fill => 'upper' 8 | 53 42
7 | 12 35 40
+-----+ 6 | 14 15 34 41
| 3| 5 | 13 11 32 39
| +--+ 4 | 10 33
| | 2| 3 | 3 8
+--+ +--+--+ 2 | 2 9 5
| 0| | | 1 | 0 7 6 28
| +--+--+ | Y=0 | 1 4
| | 1 | +------------------------------------
+-----+-----+ X=0 1 2 3 4 5 6 7 8The vibe of this is to disrupt the pattern a little bit, though the overall structure of the replications is unchanged — it's giving chaotic but structurally sound.
And just so you know: "left" is as viewed looking towards the L from above. ngl it may have been better to call it "right", but that's not changing now 🙈
🎀 Ends 🎀
Option L_fill => "ends" numbers the two endpoints within each "L", first the left then upper. This is literally the inverse of the default middle shown above, ie. it visits all the points which the middle option doesn't — so that's 2 of every 3 points in the first quadrant 💖
+-----+
| 7|
| +--+
| 6| 5|
+--+ +--+--+
| 1| 4| 2|
| +--+--+ |
| 0| 3 |
+-----+-----+
15 | 97 102 123 120 281 286 327 337
14 | 96 101 103 122 124 121 280 285 287 326 325
13 | 99 100 113 118 125 126 283 284 279 321 324
12 | 98 112 117 119 127 282 278 277 320 323
11 | 111 115 116 89 94 263 273 276 274 266
10 | 110 109 114 88 93 95 262 261 272 275 268
9 | 105 108 106 91 92 87 257 260 258 271 269
8 | 104 107 90 86 85 256 259 270 265
7 | 25 30 71 81 84 82 74 43 40
6 | 24 29 31 70 69 80 83 76 75 42 44
5 | 27 28 23 65 68 66 79 77 72 50 45
4 | 26 22 21 64 67 78 73 52 51 47
3 | 7 17 20 18 10 63 55 53 48 34
2 | 6 5 16 19 12 11 62 61 54 49 36
1 | 1 4 2 15 13 8 57 60 58 39 37
Y=0 | 0 3 14 9 56 59 38 33
+------------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14💅 All 💅
Option L_fill => "all" numbers all three points of each "L" — middle, left then right, in that order. With this one the path visits all points of the first quadrant, periodt 🔥
7 | 36 38 46 45 105 107 122 126
+-----+ 6 | 37 42 44 47 106 104 120 121
| 9 11| 5 | 41 43 33 35 98 102 103 100
| +--+ 4 | 39 40 34 32 96 97 101 99
|10| 8| 3 | 9 11 26 30 31 28 16 15
+--+ +--+--+ 2 | 10 8 24 25 29 27 19 17
| 2| 6 7| 4| 1 | 2 6 7 4 23 20 18 13
| +--+--+ | Y=0 | 0 1 5 3 21 22 14 12
| 0 1| 5 3| +--------------------------------
+-----+-----+ X=0 1 2 3 4 5 6 7Along the X=Y leading diagonal, N=0,6,24,30,96,etc — those are triples of the values from the single-point case, so 3* numbers using digits 0 and 2 in base 4, which is the same as 2* numbers using 0 and 3 in base 4. Iconic ✨
🫶 Level Ranges 🫶
For the "middles", "left" or "upper" cases with one N per tile, and taking the initial N=0 tile as level 0, a replication level is:
Nstart = 0
to
Nlevel = 4^level - 1 inclusive
Xmax = Ymax = 2 * 2^level - 1For example level 2, which is the large tiling shown in the introduction, is N=0 to N=4^2-1=15 and it extends to Xmax=Ymax=2*2^2-1=7. Cute math, honestly.
For the "ends" variation there's two points per tile, and for "all" there's three, in which case the Nlevel increases to:
Nlevel_ends = 2 * 4^level - 1
Nlevel_all = 3 * 4^level - 1⭐ FUNCTIONS ⭐
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. Bestie, that's the shared lore.
$path = Math::PlanePath::LTiling->new ()$path = Math::PlanePath::LTiling->new (L_fill => $str)— Create and return a new path object. TheL_fillchoices are:
"middle" the default
"left"
"upper"
"ends"
"all"($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. No points for negative n, literally 💀
🌸 Level Methods 🌸
($n_lo, $n_hi) = $path->level_to_n_range($level)— Return:
0, 4**$level - 1 middle, left, upper
0, 2*4**$level - 1 ends
0, 3*4**$level - 1 allThere are 4^level L shapes in a level, each containing 1, 2 or 3 points, numbered starting from 0. So many points, so much main character energy 👑
🦋 OEIS 🦋
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include:
<http://oeis.org/A062880> (etc)
L_fill=middle
A062880 N on X=Y diagonal, base 4 digits 0,2 only
A048647 permutation N at transpose Y,X
base4 digits 1<->3 and 0,2 unchanged
A112539 X+Y+1 mod 2, parity inverted
L_fill=left or upper
A112539 X+Y mod 2, parityA112539 is a parity of bits at even positions in N, ie. count 1-bits at even bit positions (least significant is bit position 0), then add 1 and take mod 2. This works because in the pattern sub-blocks 0 and 2 are unchanged and 1 and 3 are turned so as to be on opposite X,Y odd/even parity, so a flip for every even position 1-bit. L_fill=middle starts on a 0 even parity, and L_fill=left and upper start on 1 odd parity. The latter is the form in A112539 and L_fill=middle is the bitwise 0<->1 inverse. iykyk 💜
🎀 SEE ALSO 🎀
Math::PlanePath, Math::PlanePath::CornerReplicate, Math::PlanePath::SquareReplicate, Math::PlanePath::QuintetReplicate, Math::PlanePath::GosperReplicate
🌈 HOME PAGE 🌈
<http://user42.tuxfamily.org/math-planepath/index.html>
✨ LICENSE ✨
okay bestie, legal era 💅 but fr it matters, so let's get into it.
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
that's a whole decade of copyright years, we love a long-term commitment 🫶
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. 😭 (yes it's yelling in caps, that's just how licenses talk, no cap)
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 if you didn't get the license file, that link is your bestie. go get it. it's giving responsible open source citizen ✨
and that's the whole page, babe 🌸 you just read the entire Math::...Tiling(3pm) manpage, main character energy fr fr 👑
this one came from the User Contributed Perl Documentation, it's perl v5.32.0, and the page was dated 2021-01-23 🎀
k bye, go tile something iconic 💜🧚
📦 where this page comes from
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