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. A base ``L'' is replicated four times with end parts 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 | | 0 | 1 | | 0 | 1 | 7 | 4 |
+-----+ +-----+-----+ +-----+-----+-----+-----+
The parts are numbered to the left then middle then upper. This relative numbering is maintained when rotated at the next replication level, as for example N=4 to N=7.
The result is to visit 1 of every 3 points in the first quadrant with a subtle layout of 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 14
On the X=Y leading diagonal N=0,2,8,10,32,etc is the integers made from only digits 0 and 2 in base 4. Or equivalently 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 instead at their left end or upper end respectively.
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 8
The effect is to disrupt the pattern a bit though the overall structure of the replications is unchanged.
``left'' is as viewed looking towards the L from above. It may have been better to call it ``right'', but won't change that now.
Ends
Option "L_fill => "ends"" numbers the two endpoints within each ``L'', first the left then upper. This is the inverse of the default middle shown above, ie. it visits all the points which the middle option doesn't, and so 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'', 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
| 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 7
Along the X=Y leading diagonal N=0,6,24,30,96,etc 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.
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 - 1
For example level 2 which is the large tiling shown in the introduction 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 points per tile, or 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.- "$path = Math::PlanePath::LTiling->new ()"
- "$path = Math::PlanePath::LTiling->new (L_fill => $str)"
-
Create and return a new path object. The "L_fill" choices are
"middle" the default "left" "upper" "ends" "all" - "($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.
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.
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, parity
A112539 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.
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis 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/>.