NAME
Math::PlanePath::LTiling -- 2x2 self-similar of four pattern parts
SYNOPSIS
use Math::PlanePath::LTiling;
my $path = Math::PlanePath::LTiling->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This is a self-similar tiling by "L" shapes where a base L is replicated four times with the ends turned +90 and -90 degrees to make a larger L,
+-----+-----+
|12 | 15|
| +--+--+ |
| |14 | |
+--+ +--+--+
| | |11 |
| +--+ +--+
|13 | | |
+-----+ +-----+--+ +--+--+-----+
| 3 | | 3 | |10 | | 5|
| +--+ --> | +--+ +--+--+ +--+ |
| | | | | | 8 | 9 | | |
+--+ +--+--+ +--+ +--+--+--+--+ +--+
| | 2 | | | | 2 | | | 6 | |
| +--+--+ | | +--+--+ | +--+--+ |
| 0 | 1 | | 0 | 1 | 7 | 4 |
+-----+-----+ +-----+-----+-----+-----+The parts are numbered to the left then middle then upper, and this relative layout is maintained when turned for the end parts in the next replication level.
The result is to visit 1 of every 3 points in the first quadrant with a subtle layout of the points and spaces making diagonal lines and little 2x2 blocks.
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 14The N values 0,2,8,10,32,etc on the X=Y leading diagonal are all the integers made from only digits 0 and 2 in base 4. Or equivalently all integers which have zero bits at all even numbered positions (and either 0 or 1 at odd positions).
Ends
Option L_fill => "all" numbers the endpoints within each "L", the left then upper. This is the inverse of the default middle shown above and visits all the points the middle option doesn't, and on that basis 2 of every 3 points in the first quadrant.
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
+-----+
| 7|
| +--+
| 6| 5|
+--+ +--+--+
| 1| 4| 2|
| +--+--+ |
| 0| 3 |
+-----+-----+All
Option L_fill => "all" numbers all three points of each "L", as middle, left then right. With this the path visits all points of the first quadrant.
7 | 36 38 46 45 105 107 122 126
6 | 37 42 44 47 106 104 120 121
5 | 41 43 33 35 98 102 103 100
4 | 39 40 34 32 96 97 101 99
3 | 9 11 26 30 31 28 16 15
2 | 10 8 24 25 29 27 19 17
1 | 2 6 7 4 23 20 18 13
Y=0 | 0 1 5 3 21 22 14 12
+---------------------------------
X=0 1 2 3 4 5 6 7
+-----+
| 9 11|
| +--+
|10| 8|
+--+ +--+--+
| 2| 6 7| 4|
| +--+--+ |
| 0 1| 5 3|
+-----+-----+Level Ranges
For the default "middles" numbering, and taking the initial N=0 tile as level 0, a replication level is
Nstart = 0
Nlevel = 4^level - 1 inclusive
Xmax = Ymax = 2 * 2^level - 1For example level 2 which is the right hand tiling shown above is N=0 to N=4^2-1=15 and extends to Xmax=Ymax=2*2^2-1=7.
For the "ends" variation there's two per tile, or for "all" there's three, in which case the Nlevel becomes
Nlevel_ends = 2 * 4^level - 1
Nlevel_all = 3 * 4^level - 1FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::LTiling->new ()-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.
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
Copyright 2011 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/>.