Math::PlanePath::LTiling -- 2x2 self-similar of four pattern parts
use Math::PlanePath::LTiling; my $path = Math::PlanePath::LTiling->new; my ($x, $y) = $path->n_to_xy (123);
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.
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.
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
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.
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
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 $n < 0
then the return is an empty list.
($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 all
There are 4^level L shapes in a level, each containing 1, 2 or 3 points, numbered starting from 0.
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.
Math::PlanePath, Math::PlanePath::CornerReplicate, Math::PlanePath::SquareReplicate, Math::PlanePath::QuintetReplicate, Math::PlanePath::GosperReplicate
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
This file is part of Math-PlanePath.
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